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steel reinforced concrete beams
Accepted Manuscript Flexural behavior of hybrid FRP/steel reinforced concrete beams Ilker Fatih Kara, Ashraf F. Ashour, Mehmet Alpaslan Köroğlu PII: DOI: Reference:
S0263-8223(15)00266-4 http://dx.doi.org/10.1016/j.compstruct.2015.03.073 COST 6343
To appear in:
Composite Structures
Please cite this article as: Kara, I.F., Ashour, A.F., Köroğlu, M.A., Flexural behavior of hybrid FRP/steel reinforced concrete beams, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.03.073
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Flexural behavior of hybrid FRP/steel reinforced concrete beams
Ilker Fatih Karaa,b,* , Ashraf F. Ashourb and Mehmet Alpaslan Köroğlu b,c a
Civil Engineering Department, Nigde University, Nigde, Turkey
b
School of Engineering and Informatics, University of Bradford, BD7 1DP, UK
c
Civil Engineering Department, Necmettin Erbakan University, Konya, Turkey
Corresponding author, Email: [email protected], Tel: +90 388 2252293, Fax: +90 388 225 0112
Abstract This paper presents a numerical method for estimating the curvature, deflection and moment capacity of hybrid FRP/steel reinforced concrete beams.
The numerical results also indicated that beam ductility and stiffness are improved when steel reinforcement is added to FRP reinforced concrete beams. Concrete beams; Fiber reinforced polymers; Deflection; Hybrid reinforcement.
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2
A nonlinear finite element simulation for concrete beams reinforced with steel, AFRP and a hybrid combination of steel and AFRP bars [15] concluded that concrete compressive strength had a mild effect on the performance of RC beams reinforced with hybrid reinforcement. Implementing hybrid steel-FRP reinforcement has also been recognized in strengthening works. The technique known as near-surface mounting (NSM) has been adopted for the strengthening of steel-reinforced flexural beams. It is based on gluing FRP bars to the concrete near the external surfaces of the tensile zone by mounting the FRP bars in
3
groves cut in the concrete surface [16, 17]. Several experimental investigations have been reported on the bond behavior and flexural strengthening of concrete members with NSM FRP reinforcement [18-29]. Most of these studies have shown that the technique is greatly influenced by bond between FRP bars and concrete, mainly depending on FRP properties, groove surface and dimensions, adhesive and concrete properties and the position of the FRP reinforcement [21-27].
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The cross-section of reinforced concrete beams is divided into a number of concrete segments. For a particular curvature value, strains in each concrete segment and reinforcing bars are determined and stresses in each material are, hence, obtained from the respective stress-strain relationships. Equilibrium of forces and moments are eventually considered as explained below. The stress-strain relationships of concrete, steel and FRP reinforcements implemented in this investigation are shown in Figure 1. However, the numerical technique proposed can accommodate other material models. The stress-strain curve proposed by CEB-FIP [30] is adopted for concrete in compression (Figure 1(a)) and described by the following equations:
2 2 ' fc fc c c co co
f c f c'
c co
co c cu
(1(a))
(1(b))
where fc and c are the compressive stress and strain in concrete, respectively, f c' is the cylinder compressive strength of concrete, co (=0.002) is the strain in concrete at
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maximum stress and cu (=0.0035) is the ultimate strain of concrete as shown in Figure 1(a). A bi-linear stress-strain relationship is adopted to model concrete in tension as shown in Figure 1(b) and given below:
f t Et t
f t f tu
f tu
ct
t ct
t ct
((2(a))
ct (1 ) t ct
((2(b))
where ft and t are the tensile stress and strain in concrete, respectively, ftu ( 0.62 f c' ) and ct are the tensile strength and corresponding tensile strain of concrete, respectively, Et is the tensile modulus of concrete, assumed to be the same as Ec and is a factor controlling the rate of tensile strength decay. The tension stiffening effect is represented in the above model to account for concrete between cracks as it has a significant effect on member stiffness. Reinforcing steel is modelled as a bilinear elastic-perfectly plastic material with yield stress fy as shown in Figure 1(c). The stress-strain relationship of FRP bars is linear elastic up to rupture and given by:
f f Ef f
f fu
(3)
where ff and f are the stress and strain in FRP bars, respectively, Ef is the modulus of elasticity of FRP bars, and ffu and fu are the ultimate strength and strain of FRP bars, respectively, as shown in Figure 1(d).
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Figure 2 presents a concrete section reinforced with top and bottom reinforcing bars (FRP and/or steel), that is divided into a number of segments, n. The numerical analysis starts by assuming a small value of strain at the concrete extreme compression fiber. For each strain c at the top level of concrete section, the neutral axis depth, x, is initially assumed and the correct value is iteratively obtained when equilibrium of forces is satisfied. According to the assumption that plane section before bending remains plane after bending, the strain in each concrete segment is linearly proportional to its distance from the neutral axis (Figure 2(b)) as expressed below:
εi
x xi εc x
(4)
where εc is the strain at the top compression level of the reinforced concrete section and εi is the concrete compressive or tensile strain at mid-depth of i-th segment. Assuming perfect bond between concrete and reinforcing bars, strains in tensile and compressive reinforcing steel and FRP bars can also be obtained from:
' f
εf
xdf' x xdf x
εc
(5)
εc
(6)
εs
x ds εc x
(7)
's
x d s' εc x
(8)
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where f and 'f indicate the strains in bottom and top reinforcing FRP bars, s and 's are the strain in bottom and top reinforcing steel bars, df and d 'f are the bottom and top FRP reinforcement depths, respectively, and ds and d s' are the bottom and top steel reinforcement depths, respectively. The corresponding stresses in each concrete segment, and tensile and compressive reinforcements can be calculated from the respective stress-strain relationships of concrete and reinforcing bars presented in Figure 1. The total concrete force including the contribution of compressive and tensile forces is calculated using Eq. (9) below: n
Fc f ci hi b
(9)
i 1
where fci is the concrete compressive or tensile stress at the centroid of the i-th segment, hi (=h/n) is the thickness of the i-th segment, n is the total number of section segments, h is the beam depth and b is the beam width. This summation extends over all compressive and tensile concrete segments. The forces in bottom and top reinforcing bars are estimated from: Tf Af E f f
and/or
Ts As E s s
(10)
C f A'f E'f 'f
and
C s A's E's 's
(11)
Tf, Af and Ef are the force, area, and modulus of elasticity of bottom reinforcing FRP bars, respectively, Cf, A'f and E 'f are the corresponding values of top FRP reinforcement. In the same equations, Ts, As and Es are the force, area, and modulus of elasticity of bottom
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reinforcing steel bars, and Cs, A's and E 's are the corresponding values of top steel reinforcement. In calculating the compressive force of reinforcing bars Cf or Cs is used depending on the type of reinforcement placed in the compression side of concrete section. Eqs. (10) and (11) are also valid for different types of FRP bars, i.e., GFRP, AFRP and CFRP, provided that the appropriate modulus of elasticity, Ef, and tensile rupture, ffu, are used. Considering equilibrium of forces, the following equation is obtained:
Fc+C f ( s ) = T f Ts
(12a)
' ' ' f ci hi b A f ( s ) E f ( s ) f ( s ) A f E f f As E s s
(12b)
n
i 1
where Cf(s) indicates the compressive force in top FRP and/or steel reinforcing bars. In the above Eq. (12), the neutral axis depth x is the only unknown. The value of x is iteratively adjusted using the bi-section method until sufficient equilibrium accuracy is attained, for example:
Fc+C f ( s ) T f Ts Fc
108
(13)
The beam curvature φ can also be determined from the strain distribution as follows (see Figure 2(b)):
9
c x
(14)
The applied moment Mf of the section is then calculated by taking moments of internal forces about any horizontal axis, for instance about the neutral axis: n
M f Fci ( x xi ) T f ( x d f ) Ts ( x d s ) C f ( s ) ( x d f ( s ) )
(15)
i 1
where Fci (=fcihib) is the concrete compressive or tensile force at the centroid of the i-th segment. The strain in the extreme concrete compression fibre is incrementally increased and the above procedure is repeated for each value of strain until the maximum concrete compressive strain reaches its ultimate compressive strain (cu =0.0035). During such strain increase until concrete crushing, rupture of FRP bars or yielding of steel reinforcement may occur depending on the amount of hybrid FRP/steel reinforcement in the section. The section moment capacity Mfu is, therefore, the highest moment attained for various increments considered until failure. Based on the aforementioned procedure, a computer program has been developed for the moment-curvature relationship and moment capacity of FRP, steel and hybrid FRP/steel reinforced concrete sections.
2.2. Verification of the numerical technique with the experimental results 2.2.1. Flexural capacity of hybrid FRP/steel reinforced concrete beams Table 1 lists the geometrical and material properties of 46 hybrid FRP/steel reinforced concrete beams and steel-reinforced concrete beams strengthened with NSM FRP reinforcement tested in the literature. All the beams were reported to have experimentally failed in flexure. Comparisons between predictions from the current technique and flexural moment capacities of all hybrid
reinforced concrete beams collected
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are presented in Table 2 and Figure 3. The average, standard deviation and coefficient variation ratio between the present technique and experimental bending capacities, Mthe/Mexp, are 0.99, 11.3% and 11.4%, respectively. The ratio of Mthe/Mexp varies between 0.77 and 1.25. Overall, the predictions obtained from the current analysis are in good agreement with the experimental results for hybrid FRP/steel reinforced concrete beams and steel-reinforced concrete beams strengthened NSM FRP reinforcement. The failure modes are also correctly predicted by the present technique for 43 beams as reported in Table 2.
2.2.2. Moment–curvature relationship of hybrid FRP/steel reinforced concrete beams The moment-curvature results predicted from the above technique are compared with the experimental results of A1, A2 and A3 hybrid supported beams tested by
reinforced concrete simply
[2] as presented in Figure 4. Geometrical
dimensions, reinforcement details and material properties of the three beams are given in Table 1. The numerical results obtained from the present technique compare well with the test results of all the three beams as depicted in Figure 4. It also indicates that the proposed technique is able to accurately predict various features of the moment-curvature relationships, including the first cracking load, yielding and post yielding of steel, and eventually concrete crushing.
3. Effect of the amount of reinforcement on moment-curvature relationship
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The present numerical technique has been employed to study the effect of the amount of FRP or steel reinforcement on the moment-curvature relationship as presented in Figure 5: Figure 5(a) for under reinforced hybrid Figures 5(b) and (c) for over reinforced hybrid
sections, and sections. The dimensions and
concrete properties are the same for all beams considered in Figure 5; b = 200mm, h = 300mm, f c' = 35 N/mm2, ffu=600 N/mm2 and fy=480 N/mm2. The two limiting cases of pure FRP and steel reinforced concrete sections are also presented in Figure 5 against which the behaviour of hybrid FRP/steel reinforced concrete sections may be assessed. The amount of reinforcement in each case was selected such that all steel, FRP and hybrid
reinforcing bars have approximately the same tensile capacity. It can be
observed from Figure 5(a-c) that the initial linear stiffness is similar for all beams studied. The end of the linear phase is identified by concrete cracking and, consequently, reinforcing bars play a significant role in controlling the beam stiffness. After cracking, the moment-curvature relationship has a less gradient than the initial linear part owing to the decrease in the beam stiffness. However, the reduction in stiffness due to cracking varies among beams considered: the flexural stiffness after cracking is highest for steel, then hybrid FRP/steel reinforced concrete section, followed by pure FRP sections. As expected the largest stiffness reduction is observed for pure FRP sections owing to the lowest FRP elastic modulus. For the hybrid FRP/steel reinforced concrete beams, yielding of tensile steel reinforcement further reduces the beam stiffness to a similar level of pure FRP beam stiffness, followed by concrete crushing or FRP rupture. As can be seen in Fig 5(a), FRP beams exhibit the most brittle failure mode. For a , although there is a sudden drop of moment resulted from the
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rupture of FRP bars, some residual ductility is exhibited by the beam up to concrete crushing as the steel reinforcement is still far below its breaking point (Figure 5(a)). As for the over reinforced
sections, two cases of over reinforcement
configuration were considered, Figures 5(b) and (c). In the first case, Figure 5(b), while the area of FRP in the section remained constant, the steel reinforcement area was increased, whereas the area of FRP bars was increased while the steel reinforcement remained constant in the second case. In both cases, the stiffness of hybrid beams in Figures 5(b) and (c) lies between these of their counterpart steel and FRP reinforced concrete beams. As seen in Figure 5(b), the stiffness of the beams increases with the increase of steel reinforcement after the first cracking; the higher the value of As/Af, the higher the stiffness. This result could be important and beneficial for determining a suitable ratio of As/Af for hybrid beams, so that its stiffness can behave according to the required performance during the serviceability limit state. However, Figure 5(c) shows the stiffness after first cracking was similar for all the three hybrid
reinforced concrete beams though
the area of FRP bars was increased. But, the increase in the beam stiffness occurs only after yielding of steel bars. In all hybrid over reinforced beam sections, FRP bars showed an important role in resisting loads after yielding of steel while the steel reinforcement significantly contributed to the beam stiffness and load resistance after cracking. Overall, there will be ductility and stiffness improvement when steel reinforcement is added to FRP reinforced concrete beams. It is also recommended that over reinforced design be adopted to prevent excessive deformation causing brittle FRP rupture in case of under reinforced members.
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4. Effect of FRP reinforcement type on moment-curvature relationship The current numerical method has also been employed to study the effect of type of FRP reinforcement (GFRP or CRFP) on the moment-curvature relationship of hybrid reinforced concrete beams as depicted in Figure 6. The dimensions, concrete properties and amount of steel and FRP reinforcement in each case are the same for all sections considered in Figure 6; b = 200mm, h = 300mm, f c' = 35N/mm2, fy=480 N/mm2; ffu=600 N/mm2 and Ef=40 kN/mm2 for GFRP and ffu=1300 N/mm2 and Ef=120 kN/mm2 for CFRP bars. Figure 6 indicates that hybrid GFRP/steel reinforced concrete beams exhibit a significant reduction in stiffness after yielding of steel in comparison with hybrid CFRP/steel reinforced concrete section. This behavior is mainly attributed to the lower elastic modulus of GFRP bars than that of CFRP bars leading to a reduced effective moment of inertia and hence larger curvature. It can be also observed from Figure 6 that increasing the tensile FRP reinforcement slightly increases the flexural stiffness of either CFRP/steel or GFRP/steel concrete members, especially after yielding of steel.
5. Failure modes of hybrid FRP/steel reinforced concrete beams For concrete beams reinforced with either steel or FRP, the balanced reinforcement ratio is attained when the beam fails by crushing of concrete in compression and either yielding of steel or rupture of FRP in tension, simultaneously. However, a failure condition of hybrid FRP/steel reinforced concrete beams, in which crushing of concrete, yielding of steel and rupture of FRP simultaneously take place, is almost impossible to occur in practice. The steel reinforcement would have yielded long before rupture of FRP
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reinforcement as the steel yield strain (y ≈0.002) is far much less than the FRP rupture strain (f ≈0.01). Hence, the balanced condition for hybrid FRP/steel reinforced concrete beams is proposed in a way such that concrete crushing in compression and rupture of FRP reinforcement simultaneously occur, while steel reinforcement would have already yielded [12]. Figure 7 presents the strain distribution for various modes of failure as explained below in more details. 5.1 Mode I: Tensile rupture of FRP bars When εf =εfu and εc =εcu simultaneously occur, the strain distribution is unique and the neutral axis depth xl is calculated using Eq. (6) with εf and εc replaced by εfu and 0.0035, respectively:
0.0035 0.0035 fu
xl d f
(16)
When the area Af of FRP reinforcement is below a certain value Afl, FRP brittle rupture is likely to occur before concrete crushing. For a given area As of steel reinforcement, the limiting area Afl of FRP bars can be readily calculated from Eq. 12(b) with fs and ff replaced by fy and ffu, respectively:
n
Afl
f i 1
ci
bhi A'f ( s ) E 'f ( s ) 'f ( s ) - As f y (17)
f fu
where ε'f(s) is calculated from Eq. (8) with x and εc replaced by xl and 0·0035, respectively. The area Afl of FRP bars (given by Eq. (17)) forms a lower limit for the amount of FRP bars in order to avoid tensile FRP rupture. Figure 7(a) shows the strain distribution in case of Af < Afl.
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5.2 Mode II: Yielding of steel reinforcement followed by crushing of concrete This mode of failure is characterized by yielding of steel reinforcement followed by crushing of concrete, before rupture of FRP reinforcement (see Figure 7(b) for strain distribution). In this case, the area of FRP bars is greater than Afl, therefore tensile rupture of FRP bars would not occur. This case is similar to the under-reinforced section of conventional reinforced concrete beams. The upper limit to the area of FRP bars, Afu, is reached when strains εc in the extreme compression fibre of concrete and εs in the tension reinforcement simultaneously reach εcu and εy, respectively. By following similar calculations to those presented above for mode I, the neutral axis depth xu and the area Afu of FRP bars corresponding to this upper limit of FRP bars are calculated from:
xu d s
0.0035 0.0035 y n
Afu
f i 1
ci
(18)
bhi A'f ( s ) E 'f ( s ) 'f ( s ) - As f y (19)
Ef f
where ε'f(s) and εf are calculated from Eqs. (6) and (8), respectively, with x and εc replaced by xu and 0·0035, respectively. The neutral axis depth xu given by Eq. (18) is the same as that defining the balanced section for conventional reinforced concrete beams. To ensure ductile flexural behavior, the area of FRP bars provided should be smaller than Afu given in Eq. (19) and greater than Afl to achieve ductile failure by yielding of steel reinforcement and crushing of concrete, before rupture of FRP reinforcement.
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5.3 Mode III: Crushing of concrete before yielding of steel reinforcement (overreinforced case) If the area of FRP bars used is greater than Afu, the concrete strain εc reaches the ultimate value εcu before any yielding of tension reinforcement or FRP rupture as shown in Figure 7(c). In this case, there is a large amount of steel and FRP bars and the section is overreinforced. Such failure, often explosive, occurs with little warning, similar to that of conventional over reinforced concrete beams.
5.4 Effect of size of steel and FRP bars on the bending capacity Figure 8 presents the effect of steel and FRP reinforcements on the normalized moment capacity η (= Mf/bd2): Figure 8(a) shows the variation of the normalized moment capacity η against the steel reinforcement ratio ρs (= 100As/bd) whereas Figure 8(b) gives the variation of the normalized moment capacity η against the FRP reinforcement ratio ρf (= 100Af/bd). The dotted lines in Figure 8 represent the boundaries of different flexural failure modes as explained above. Overall, the effect of FRP reinforcement is more pronounced for hybrid FRP/steel reinforced concrete sections having less area of steel reinforcement, especially when mode II dominates the flexural behaviour of hybrid reinforced concrete beams. However, the increase in the normalized moment capacity η with the increase of hybrid FRP/steel reinforcement is insignificant when mode III (crushing of concrete before yielding of steel and rupture of FRP) dominates the flexural failure. The higher the amount of FRP bars, the less the rate of increase of the normalized moment capacity η for the same steel reinforcement ratio ρs. Therefore, it is recommended that the upper limit to the area of FRP reinforcement, Afu given by (Eq.
17
(19)), be selected as it shows optimum use of FRP reinforcement and exhibits more ductility.
6. Prediction of deflections A simplified method for deflection calculation of hybrid FRP/steel reinforced concrete beams is presented from the moment-curvature relationship developed above. The beam flexural rigidity, EIeff, corresponding to the maximum moment along the beam is firstly determined from the moment-curvature relationship at each loading as in Eq. (20) below:
EI eff
M
(20)
The midspan deflection, of hybrid FRP/steel reinforced concrete beams is then calculated using the elastic deflection formula for the loading pattern, for example, the mid-span deflection of simply supported beams loaded with a mid-span point load, P, could be calculated from Eq. (21) below:
PL3 48 EI eff
(21)
On the other hand, the immediate deflection of simply supported beams loaded with twopoint loads, each P/2, could be also computed from Eq. (22) as follows:
( P / 2)(3L2 4a 2 ) 24 EI eff
(22)
where a is the shear span and L is the span length. Bischoff [31] recently developed the following effective moment of inertia equation that showed a good agreement with test
18
results for the prediction of deflections of either steel or FRP reinforced concrete beams and also hybrid FRP/steel reinforced concrete beams up to the serviceability loading level [10].
I eff
I2 2 1 1 I 2 M cr I M 1
(23)
where Mcr is the flexural cracking moment, M is the applied bending moment, I1 (=bh3 /12) is the gross moment of inertia and I2 is the moment of inertia of transformed cracked section considering the contribution of steel and FRP reinforcement to the stiffness, Mcr (=2fcrI1 /h) is the cracking moment, fcr (= 0.62 f c ) is the modulus of rupture of concrete. The mid-span experimental deflections of six hybrid FRP/steel reinforced concrete simply supported beams tested by Lau and Pand [4] and Qu et al. [5] are compared with the predictions from the current numerical technique and Bischoff’s equation; see Figure 9(a-f). The geometrical dimensions, reinforcement details and material properties of hybrid FRP/steel reinforced concrete beams considered are given in Table 1. The deflection results obtained from the present numerical technique are in good agreement with the test results for the applied loads up to failure. Figure 9 indicates that the present technique is able to predict both the pre and post cracking deflections and to capture the beam stiffness and deflection after yielding of steel. Figure 9 also shows that the deflections predicted by Bischoff model compare well at low load levels before and after
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the first cracking load. However, for high load levels, the predictions of Bischoff model significantly underestimate the experimental results. Fig. 10 shows the variation of the flexural rigidity, EIeff, of three hybrid beams against the applied moment. The moment-effective flexural rigidity curve consists of different segments, denoting the pre-cracking, post-cracking, yielding of steel, FRP rupture and concrete crushing stages. In the pre-cracking stage, the beams showed the highest flexural stiffness that is gradually decreased with the increase in the applied moment. Once the beams cracked, the flexural rigidity was notably reduced but remained almost constant until yielding of steel reinforcement. A further reduction of EIeff is depicted after the yielding of tensile steel reinforcement up to concrete crashing failure in case of over reinforced beams or FRP rupture in case of under reinforced beams.
7. Conclusions A numerical technique based on equilibrium of forces and full compatibility of strains has been developed to evaluate the moment-curvature relationship, moment capacity and deflections of hybrid FRP/steel reinforced concrete beams and
reinforced concrete
beams strengthened with NSM FRP bars/strips. Comparisons between experimental results available in the literature and predicted curvature, moment capacity and deflection of hybrid FRP/steel reinforced concrete beams show good agreement. The proposed model also gives good results for the moment capacity of reinforced concrete beams strengthened with NSM FRP bars/strips. The proposed technique is able to predict both the pre and post cracking deflections up to the beam failure and to capture the beam behaviour after yielding of steel or rupture of FRP reinforcements. The Bischoff model
20
for deflection prediction compares well at low load levels before and after the first cracking load, but, significantly underestimates the experimental results for high load levels. A parametric study concluded that the contribution of steel reinforcement to FRP concrete beams provides ductility and stiffness improvement of beams studied. The FRP reinforcement was found to play an important role to resist loading after yielding of steel in over reinforced sections. The results also indicated that hybrid GFRP/steel reinforced concrete beams exhibit a significant reduction in stiffness after the initiation of first crack and yielding of steel reinforcement in comparison with hybrid CFRP/steel concrete beams.
ACKNOWLEDGMENT The first author acknowledges the financial support of the Scientific and Technological Research Council of Turkey (TUBİTAK).
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[31] Bischoff PH, Deflection calculation of FRP reinforced concrete beams based on modifications to the existing Branson equation, J Compos Constr, 2007, 11(1), 4– 14.
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ft
fc Parabolic curve
ftu
f c'
Ec
co
Et
c
cu
(a) Concrete in compression
ct
fs
(b) Concrete in tension ff
fy
ffu
s
Es
y
t
μct
f
Ef
fu (d) FRP composites
(c) Steel
Fig. 1 FRP, steel and concrete stress–strain relationships.
26
d'f(s)
εc
b A'f(s)
xi
h
df(s)
ε'f(s)
x Neutral axis
Cf(s) Fc
εi
segment i Concrete tensile stress
Af + As
(a) hybrid reinforced concrete section
εf and εs (b) Strain distribution
Concrete compressive stress
Tf+ Ts
(c) Stress distribution and forces
Fig. 2 Strains, stresses and forces of concrete section reinforced with FRP/steel bars
27
300 250
Mth(kNm)
200 150 100 50 0 0
50
100
150
200
250
300
Mexp(kNm)
Fig. 3 Experimental versus predicted moment capacities of hybrid FRP/steel reinforced concrete beams
28
Fig. 4 Comparison of predicted and experimental moment-curvature relationships of three hybrid FRP/steel reinforced concrete beams.
29
60
Moment(kNm)
45
30
15
FRP R/C Beam(Af=307 mm2) Steel R/C Beam(As=402 mm2) Hybrid R/C Beam (As=157 mm2, Af=307 mm2)
0 0
0.03
0.06
0.09
Curvature (1/m)
(a) Under reinforced sections
100
Moment(kNm)
80 60 40 Hybrid R/C Beam (As=402, Af=402mm2) Hybrid R/C Beam (As=509, Af=402mm2) Steel R/C Beam(As=763 mm2) FRP R/C Beam(Af=763 mm2) Hybrid R/C Beam (As=450, Af=402 mm2)
20 0 0
0.01
0.02
0.03
0.04
0.05
0.06
Curvature (1/m)
(b) Over reinforced hybrid FRP/steel sections (Af remained constant in hybrid FRP/steel sections)
30
100
Moment(kNm)
80 60 40 Hybrid R/C Beam (As=402, Af=307mm2) Hybrid R/C Beam (As=402, Af=402mm2) Steel R/C Beam (As=763 mm2) FRP R/C Beam (Af=763 mm2) Hybrid R/C Beam (As=402, Af=509mm2)
20 0 0
0.01
0.02
0.03
0.04
0.05
0.06
Curvature (1/m)
(c) Over reinforced hybrid FRPRC sections (As remained constant in hybrid FRP/steel sections) Fig. 5 (a-c) Moment-curvature relationships of under and over reinforced concrete sections
31
Fig. 6 Effect of type of FRP reinforcement (GFRP or CFRP) on the moment curvature relationships of hybrid FRP/steel reinforced concrete beam sections
32
x≤xl
ε'f(s)
xl
εs> εsy
εf= εfu (a)
εc=0.0035
εc=0.0035
εc≤ 0.0035
ε'f(s)
x>xu
εs≥ εsy
εf< εfu (b)
ε'f(s)
εs< εsy
εf< εfu (c)
Fig. 7 Strain distribution for different amount of FRP bars at failure (a) FRP rupture before concrete crushing (Af ≤ Afl) (b) Case of (Afl < Af ≤ Afu) (c) Over-reinforced case (Af > Afu)
33
10 Boundary of over-reinforced case
9 8
ρf=4 ρf=3
η=Mf /bd2
7 6
Ef=40 kN/mm2 ffu=780 N/mm2 bw=180 mm h= 280 mm df=250 mm ds=220mm
ρf=2
5
ρf=1
4
ρf=0.5
3 ρf=0.2 ρf=0.1
2 1
Boundary of GFRP rupture
ρf=0
0 0
0.5
1
1.5
2
2.5
3
3.5
4
100 A s /bd
(a) Variation of moment capacity against the area of steel reinforcement
9
Boundary of over-reinforced case
ρs=4
8 ρs=3
7 6
η=Mf /bd
2
ρs=2
5 ρs=1
4
Ef=40 kN/mm2 ffu=780 N/mm2 bw=180 mm h= 280 mm df=250 mm ds=220mm
3 2 ρs=0
Boundary of GFRP rupture
1 0 0
0.5
1
1.5
2
2.5
3
100 A f /bd
(b) Variation of moment capacity against the area of FRP bar Fig. 8 (a-b) Effect of steel and FRP reinforcements on the normalised moment capacity 160 140 (kN)
120 100
34
250
Load(kN)
200 150 100 G06T1-Numerical G06T1-Experimental Bischoff's equation
50 0 0
20
40
60
80
Midspan deflection(mm)
(b)
35
100
120
300 250 Load(kN)
200 150 100 G1T07-Numerical G1T07-Experimental Bischoff's equation
50 0 0
20
40
60
80
100
120
Midspan deflection(mm)
(c)
160
Load(kN)
120 80 40
B5-Numerical B5-Experimental B5-Bischoff's equation
0 0
5
10
15
20
Midspan deflection(mm)
(d)
36
25
30
200
120 80 B6-Numerical B6-Experimental B6-Bischoff's equation
40 0 0
5
10
15
20
25
Midspan deflection(mm)
(e)
120
Load(kN)
Load(kN)
160
80
40 B7-Numerical B7-Experimental B7-Bischoff's equation
0 0
5
10
15
20
25
30
35
40
Midspan deflection(mm)
(f) Fig. 9 (a-f) Comparison between experimental and theoretical deflections of 6 hybrid FRP/steel reinforced concrete beams tested in the literature
37
concrete cracking
steel yielding point
FRP rupture
Fig. 10 Effective flexural stiffness variations
38
Table 1. Details of hybrid steel/FRP reinforced concrete beams tested in the literature. Referen Beam ce notatio n [1]
A2
A3
B2
[2]
A1
88.3 A2
157.0 A3
C1
[10]
B3
B4
Type of loadin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g 49.0
Wid th (mm ) 175
Over all depth (mm) 350
Spa n (m m) 230 0
f c' (MP a)
Af (mm2 )
As Ef 2 (mm (kN/m ) m2)
Type of FRP
31.0
50.2
150. 7
52.0
AFR P
175
350
230 0
30.9
25.1
301. 4
52.0
AFR P
200
400
230 0
30.0
25.1
150. 7
52.0
AFR P
150
200
270 0
38.8
100. 5
270 0
38.8
100. 5
200
270 0
38.8
226. 1
235. 5
50.1
AFR P
200
270 0
38.8
100. 5
88.3
49.0
AFR P
250
180 0
28.1
226. 1
253. 2
45.0
GFR P
250
180 0
28.1
201. 0
396. 9
41.0
GFR P
AFR P 150 200
Four point bendin g 50.1 AFR P Four 150 point bendin g Four 150 point bendin g Four 180 point bendin g Four 180 point bendin g
39
B5
B6
B7
B8
[11]
L2
L5
H2
[12]
G03M D1
G1T07
G06T1
[13]
CS
GS
Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Three point bendin g Three point bendin g Three point bendin g Four point bendin g Four point
180
250
180 0
29.2
401. 9
141. 7
37.7
GFR P
180
250
180 0
29.2
401. 9
253. 2
45.0
GFR P
180
250
180 0
34.6
113. 0
141. 7
37.7
GFR P
180
250
180 0
34.6
1205 .8
396. 9
41.0
GFR P
150
200
220 0
28.5
141. 7
157. 0
40.8
GFR P
150
200
220 0
28.5
212. 5
157. 0
40.8
GFR P
150
200
220 0
48.5
141. 8
157. 0
40.8
GFR P
280
380
420 0
41.3
283. 5
981. 7
39.5
GFR P
280
380
420 0
39.8
981. 7
628. 3
38.0
GFR P
280
380
420 0
44.6
567. 1
981. 7
39.5
GFR P
230
250
190 0
75.9
128. 0
485. 0
146.2
CFR P
230
250
190 0
75.9
381. 0
485. 0
48.1
GFR P
40
[14]
B10/6S
B10/8S
B12/6S
B12/8S
[18]
RW1F
RW2F
[19]
ALII-2
AMI-2
AMII2
AMIII -1
AMIII -2
bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng
100
200
122 0
30.0
56.6
157. 0
41.0
GFR P
100
200
122 0
30.0
100. 6
157. 0
39.0
GFR P
100
200
122 0
30.0
56.6
226. 0
41.0
GFR P
100
200
122 0
30.0
100. 6
226. 0
39.0
GFR P
150
200
200 0
36.6
78.5
157
40
GFR P
150
200
200 0
36.6
157
78.5
40
GFR P
250
400
360 0
27
56
595. 37
165
CFR P
250
400
360 0
27
56
379. 84
165
CFR P
250
400
360 0
27
56
595. 37
165
CFR P
250
400
360 0
27
28
859. 12
165
CFR P
250
400
360 0
27
56
859. 12
165
CFR P
41
AMIII -3
[20]
[21]
[28]
Four point bendi ng BMIV- Four 1 point bendi ng BMIV- Four 2 point bendi ng BMIV- Four 3 point bendi ng 6-1Fa Four point bendi ng 6-2Fa Four point bendi ng 9-1Fa Four point bendi ng 9-2Fa Four point bendi ng 12-1Fa Four point bendi ng 12-2Fa Four point bendi ng SC-12 Four point bendi ng Nonpr Four s point
250
400
360 0
27
84
859. 12
165
CFR P
400
200
360 0
27
28
859. 12
165
CFR P
400
200
360 0
27
56
859. 12
165
CFR P
400
200
360 0
27
84
859. 12
165
CFR P
152. 4
190
274 3
37.2
37.5
402
136
CFR P
152. 4
190
274 3
37.2
75
402
136
CFR P
229
190
274 3
37.2
37.5
398
136
CFR P
229
190
274 3
37.2
75
398
136
CFR P
305
190
274 3
37.2
37.5
398
136
CFR P
305
190
274 3
37.2
75
398
136
CFR P
150
280
280 0
35.1
113
226
146
CFR P
152
254
330 0
47.9
70.8 5
196. 94
136
CFR P
42
[29]
BM1
BM2
bendi ng Four point bendi ng Four point bendi ng
200
300
360 0
64
100
402
250
CFR P
200
300
360 0
65
100
402
250
CFR P
Note: Ef is the modulus of elasticity of FRP longitudinal bars, Af and As are FRP and steel reinforcement areas and f c' is the cylinder compressive strength of concrete. If the concrete cylinder compressive strength f c' is not measured in tests, it is assumed that f c' 0.85 f cu , where
f cu is the concrete cube compressive strength.
43
Table 2. Comparisons between the theoretical and experimental flexural moment capacities of hybrid steel/FRP reinforced concrete beams tested in the literature.
Reference [1]
[2]
[10]
[11]
[12]
[13] [14]
[18] [19]
[20]
Beam notation A2 A3 B2 A1 A2 A3 C1 B3 B4 B5 B6 B7 B8 L2 L5 H2 G03MD1 G1T07 G06T1 CS GS B10/6S B10/8S B12/6S B12/8S RW1F RW2F ALII-2 AMI-2 AMII-2 AMIII-1 AMIII-2 AMIII-3 BMIV-1 BMIV-2 BMIV-3 6-1Fa 6-2Fa 9-1Fa
Mexp (kN) 36.00 43.50 35.25 25.14 28.41 35.55 25.14 38.38 39.66 36.36 42.57 23.55 63.30 22.23 23.07 21.11 147.00 261.00 229.00 78.20 82.80 14.09 14.41 14.89 16.33 21.84 23.03 145.9 110.15 147.62 156.36 174.08 180.44 66.77 71.09 78.69 30.37 33.06 34.13
Mthe (kN)
34.42 43.79 28.16 19.93 25.49 33.24 25.10 38.17 42.64 42.36 51.74 29.39 78.50 17.17 19.51 19.56 141.78 231.60 245.11 67.49 77.28 13.63 15.16 15.09 16.37 17.62 19.29 147.34 117.89 147.46 166.91 185.48 200.65 66.78 73.22 78.68 31.37 35.17 33.54
44
Mthe / Mexp 0.96 1.00 0.80 0.79 0.90 0.94 1.00 0.99 1.07 1.17 1.21 1.25 1.24 0.77 0.85 0.93 0.97 0.89 1.07 0.86 0.93 0.97 0.95 0.99 1.00 0.81 0.84 1.01 1.07 1.00 1.07 1.07 1.11 1.00 1.03 1.00 1.03 1.06 0.98
Experimental mode of failure SY-FRPR SY-FRPR SY-FRPR SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-FRPR SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-FRPR SY-FRPR SY-CC SY-FRPR SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC CC CC CC*
[21] [28] [29]
9-2Fa 12-1Fa 12-2Fa SC-12 Nonprs BM1 BM2
44.37 35.83 41.08 64.4 53.08 79.3 79.3
38.69 34.78 43.75 68.8 54.32 87.3 85.1 Average Standard deviation (%) Coefficient of variation (%)
0.87 0.97 1.06 1.07 1.02 1.1 1.07 0.99 11.4%
CC CC* CC* SY-CC SY-FRPR SY-FRPR SY-FRPR
11.51%
Note: Mexp and Mthe are the experimental and theoretical moment capacities of hybrid steel/FRP sections. Experimental failure modes: SY-CC refers to steel yielding followed by concrete crushing and SY-FRPR indicates steel yielding followed by FRP rupture modes of failure, CC refers to concrete crushing. *Indicates disagreement between predicted and experimentally observed flexural failure modes.
45
S0263-8223(15)00266-4 http://dx.doi.org/10.1016/j.compstruct.2015.03.073 COST 6343
To appear in:
Composite Structures
Please cite this article as: Kara, I.F., Ashour, A.F., Köroğlu, M.A., Flexural behavior of hybrid FRP/steel reinforced concrete beams, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.03.073
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Flexural behavior of hybrid FRP/steel reinforced concrete beams
Ilker Fatih Karaa,b,* , Ashraf F. Ashourb and Mehmet Alpaslan Köroğlu b,c a
Civil Engineering Department, Nigde University, Nigde, Turkey
b
School of Engineering and Informatics, University of Bradford, BD7 1DP, UK
c
Civil Engineering Department, Necmettin Erbakan University, Konya, Turkey
Corresponding author, Email: [email protected], Tel: +90 388 2252293, Fax: +90 388 225 0112
Abstract This paper presents a numerical method for estimating the curvature, deflection and moment capacity of hybrid FRP/steel reinforced concrete beams.
The numerical results also indicated that beam ductility and stiffness are improved when steel reinforcement is added to FRP reinforced concrete beams. Concrete beams; Fiber reinforced polymers; Deflection; Hybrid reinforcement.
1
2
A nonlinear finite element simulation for concrete beams reinforced with steel, AFRP and a hybrid combination of steel and AFRP bars [15] concluded that concrete compressive strength had a mild effect on the performance of RC beams reinforced with hybrid reinforcement. Implementing hybrid steel-FRP reinforcement has also been recognized in strengthening works. The technique known as near-surface mounting (NSM) has been adopted for the strengthening of steel-reinforced flexural beams. It is based on gluing FRP bars to the concrete near the external surfaces of the tensile zone by mounting the FRP bars in
3
groves cut in the concrete surface [16, 17]. Several experimental investigations have been reported on the bond behavior and flexural strengthening of concrete members with NSM FRP reinforcement [18-29]. Most of these studies have shown that the technique is greatly influenced by bond between FRP bars and concrete, mainly depending on FRP properties, groove surface and dimensions, adhesive and concrete properties and the position of the FRP reinforcement [21-27].
4
The cross-section of reinforced concrete beams is divided into a number of concrete segments. For a particular curvature value, strains in each concrete segment and reinforcing bars are determined and stresses in each material are, hence, obtained from the respective stress-strain relationships. Equilibrium of forces and moments are eventually considered as explained below. The stress-strain relationships of concrete, steel and FRP reinforcements implemented in this investigation are shown in Figure 1. However, the numerical technique proposed can accommodate other material models. The stress-strain curve proposed by CEB-FIP [30] is adopted for concrete in compression (Figure 1(a)) and described by the following equations:
2 2 ' fc fc c c co co
f c f c'
c co
co c cu
(1(a))
(1(b))
where fc and c are the compressive stress and strain in concrete, respectively, f c' is the cylinder compressive strength of concrete, co (=0.002) is the strain in concrete at
5
maximum stress and cu (=0.0035) is the ultimate strain of concrete as shown in Figure 1(a). A bi-linear stress-strain relationship is adopted to model concrete in tension as shown in Figure 1(b) and given below:
f t Et t
f t f tu
f tu
ct
t ct
t ct
((2(a))
ct (1 ) t ct
((2(b))
where ft and t are the tensile stress and strain in concrete, respectively, ftu ( 0.62 f c' ) and ct are the tensile strength and corresponding tensile strain of concrete, respectively, Et is the tensile modulus of concrete, assumed to be the same as Ec and is a factor controlling the rate of tensile strength decay. The tension stiffening effect is represented in the above model to account for concrete between cracks as it has a significant effect on member stiffness. Reinforcing steel is modelled as a bilinear elastic-perfectly plastic material with yield stress fy as shown in Figure 1(c). The stress-strain relationship of FRP bars is linear elastic up to rupture and given by:
f f Ef f
f fu
(3)
where ff and f are the stress and strain in FRP bars, respectively, Ef is the modulus of elasticity of FRP bars, and ffu and fu are the ultimate strength and strain of FRP bars, respectively, as shown in Figure 1(d).
6
Figure 2 presents a concrete section reinforced with top and bottom reinforcing bars (FRP and/or steel), that is divided into a number of segments, n. The numerical analysis starts by assuming a small value of strain at the concrete extreme compression fiber. For each strain c at the top level of concrete section, the neutral axis depth, x, is initially assumed and the correct value is iteratively obtained when equilibrium of forces is satisfied. According to the assumption that plane section before bending remains plane after bending, the strain in each concrete segment is linearly proportional to its distance from the neutral axis (Figure 2(b)) as expressed below:
εi
x xi εc x
(4)
where εc is the strain at the top compression level of the reinforced concrete section and εi is the concrete compressive or tensile strain at mid-depth of i-th segment. Assuming perfect bond between concrete and reinforcing bars, strains in tensile and compressive reinforcing steel and FRP bars can also be obtained from:
' f
εf
xdf' x xdf x
εc
(5)
εc
(6)
εs
x ds εc x
(7)
's
x d s' εc x
(8)
7
where f and 'f indicate the strains in bottom and top reinforcing FRP bars, s and 's are the strain in bottom and top reinforcing steel bars, df and d 'f are the bottom and top FRP reinforcement depths, respectively, and ds and d s' are the bottom and top steel reinforcement depths, respectively. The corresponding stresses in each concrete segment, and tensile and compressive reinforcements can be calculated from the respective stress-strain relationships of concrete and reinforcing bars presented in Figure 1. The total concrete force including the contribution of compressive and tensile forces is calculated using Eq. (9) below: n
Fc f ci hi b
(9)
i 1
where fci is the concrete compressive or tensile stress at the centroid of the i-th segment, hi (=h/n) is the thickness of the i-th segment, n is the total number of section segments, h is the beam depth and b is the beam width. This summation extends over all compressive and tensile concrete segments. The forces in bottom and top reinforcing bars are estimated from: Tf Af E f f
and/or
Ts As E s s
(10)
C f A'f E'f 'f
and
C s A's E's 's
(11)
Tf, Af and Ef are the force, area, and modulus of elasticity of bottom reinforcing FRP bars, respectively, Cf, A'f and E 'f are the corresponding values of top FRP reinforcement. In the same equations, Ts, As and Es are the force, area, and modulus of elasticity of bottom
8
reinforcing steel bars, and Cs, A's and E 's are the corresponding values of top steel reinforcement. In calculating the compressive force of reinforcing bars Cf or Cs is used depending on the type of reinforcement placed in the compression side of concrete section. Eqs. (10) and (11) are also valid for different types of FRP bars, i.e., GFRP, AFRP and CFRP, provided that the appropriate modulus of elasticity, Ef, and tensile rupture, ffu, are used. Considering equilibrium of forces, the following equation is obtained:
Fc+C f ( s ) = T f Ts
(12a)
' ' ' f ci hi b A f ( s ) E f ( s ) f ( s ) A f E f f As E s s
(12b)
n
i 1
where Cf(s) indicates the compressive force in top FRP and/or steel reinforcing bars. In the above Eq. (12), the neutral axis depth x is the only unknown. The value of x is iteratively adjusted using the bi-section method until sufficient equilibrium accuracy is attained, for example:
Fc+C f ( s ) T f Ts Fc
108
(13)
The beam curvature φ can also be determined from the strain distribution as follows (see Figure 2(b)):
9
c x
(14)
The applied moment Mf of the section is then calculated by taking moments of internal forces about any horizontal axis, for instance about the neutral axis: n
M f Fci ( x xi ) T f ( x d f ) Ts ( x d s ) C f ( s ) ( x d f ( s ) )
(15)
i 1
where Fci (=fcihib) is the concrete compressive or tensile force at the centroid of the i-th segment. The strain in the extreme concrete compression fibre is incrementally increased and the above procedure is repeated for each value of strain until the maximum concrete compressive strain reaches its ultimate compressive strain (cu =0.0035). During such strain increase until concrete crushing, rupture of FRP bars or yielding of steel reinforcement may occur depending on the amount of hybrid FRP/steel reinforcement in the section. The section moment capacity Mfu is, therefore, the highest moment attained for various increments considered until failure. Based on the aforementioned procedure, a computer program has been developed for the moment-curvature relationship and moment capacity of FRP, steel and hybrid FRP/steel reinforced concrete sections.
2.2. Verification of the numerical technique with the experimental results 2.2.1. Flexural capacity of hybrid FRP/steel reinforced concrete beams Table 1 lists the geometrical and material properties of 46 hybrid FRP/steel reinforced concrete beams and steel-reinforced concrete beams strengthened with NSM FRP reinforcement tested in the literature. All the beams were reported to have experimentally failed in flexure. Comparisons between predictions from the current technique and flexural moment capacities of all hybrid
reinforced concrete beams collected
10
are presented in Table 2 and Figure 3. The average, standard deviation and coefficient variation ratio between the present technique and experimental bending capacities, Mthe/Mexp, are 0.99, 11.3% and 11.4%, respectively. The ratio of Mthe/Mexp varies between 0.77 and 1.25. Overall, the predictions obtained from the current analysis are in good agreement with the experimental results for hybrid FRP/steel reinforced concrete beams and steel-reinforced concrete beams strengthened NSM FRP reinforcement. The failure modes are also correctly predicted by the present technique for 43 beams as reported in Table 2.
2.2.2. Moment–curvature relationship of hybrid FRP/steel reinforced concrete beams The moment-curvature results predicted from the above technique are compared with the experimental results of A1, A2 and A3 hybrid supported beams tested by
reinforced concrete simply
[2] as presented in Figure 4. Geometrical
dimensions, reinforcement details and material properties of the three beams are given in Table 1. The numerical results obtained from the present technique compare well with the test results of all the three beams as depicted in Figure 4. It also indicates that the proposed technique is able to accurately predict various features of the moment-curvature relationships, including the first cracking load, yielding and post yielding of steel, and eventually concrete crushing.
3. Effect of the amount of reinforcement on moment-curvature relationship
11
The present numerical technique has been employed to study the effect of the amount of FRP or steel reinforcement on the moment-curvature relationship as presented in Figure 5: Figure 5(a) for under reinforced hybrid Figures 5(b) and (c) for over reinforced hybrid
sections, and sections. The dimensions and
concrete properties are the same for all beams considered in Figure 5; b = 200mm, h = 300mm, f c' = 35 N/mm2, ffu=600 N/mm2 and fy=480 N/mm2. The two limiting cases of pure FRP and steel reinforced concrete sections are also presented in Figure 5 against which the behaviour of hybrid FRP/steel reinforced concrete sections may be assessed. The amount of reinforcement in each case was selected such that all steel, FRP and hybrid
reinforcing bars have approximately the same tensile capacity. It can be
observed from Figure 5(a-c) that the initial linear stiffness is similar for all beams studied. The end of the linear phase is identified by concrete cracking and, consequently, reinforcing bars play a significant role in controlling the beam stiffness. After cracking, the moment-curvature relationship has a less gradient than the initial linear part owing to the decrease in the beam stiffness. However, the reduction in stiffness due to cracking varies among beams considered: the flexural stiffness after cracking is highest for steel, then hybrid FRP/steel reinforced concrete section, followed by pure FRP sections. As expected the largest stiffness reduction is observed for pure FRP sections owing to the lowest FRP elastic modulus. For the hybrid FRP/steel reinforced concrete beams, yielding of tensile steel reinforcement further reduces the beam stiffness to a similar level of pure FRP beam stiffness, followed by concrete crushing or FRP rupture. As can be seen in Fig 5(a), FRP beams exhibit the most brittle failure mode. For a , although there is a sudden drop of moment resulted from the
12
rupture of FRP bars, some residual ductility is exhibited by the beam up to concrete crushing as the steel reinforcement is still far below its breaking point (Figure 5(a)). As for the over reinforced
sections, two cases of over reinforcement
configuration were considered, Figures 5(b) and (c). In the first case, Figure 5(b), while the area of FRP in the section remained constant, the steel reinforcement area was increased, whereas the area of FRP bars was increased while the steel reinforcement remained constant in the second case. In both cases, the stiffness of hybrid beams in Figures 5(b) and (c) lies between these of their counterpart steel and FRP reinforced concrete beams. As seen in Figure 5(b), the stiffness of the beams increases with the increase of steel reinforcement after the first cracking; the higher the value of As/Af, the higher the stiffness. This result could be important and beneficial for determining a suitable ratio of As/Af for hybrid beams, so that its stiffness can behave according to the required performance during the serviceability limit state. However, Figure 5(c) shows the stiffness after first cracking was similar for all the three hybrid
reinforced concrete beams though
the area of FRP bars was increased. But, the increase in the beam stiffness occurs only after yielding of steel bars. In all hybrid over reinforced beam sections, FRP bars showed an important role in resisting loads after yielding of steel while the steel reinforcement significantly contributed to the beam stiffness and load resistance after cracking. Overall, there will be ductility and stiffness improvement when steel reinforcement is added to FRP reinforced concrete beams. It is also recommended that over reinforced design be adopted to prevent excessive deformation causing brittle FRP rupture in case of under reinforced members.
13
4. Effect of FRP reinforcement type on moment-curvature relationship The current numerical method has also been employed to study the effect of type of FRP reinforcement (GFRP or CRFP) on the moment-curvature relationship of hybrid reinforced concrete beams as depicted in Figure 6. The dimensions, concrete properties and amount of steel and FRP reinforcement in each case are the same for all sections considered in Figure 6; b = 200mm, h = 300mm, f c' = 35N/mm2, fy=480 N/mm2; ffu=600 N/mm2 and Ef=40 kN/mm2 for GFRP and ffu=1300 N/mm2 and Ef=120 kN/mm2 for CFRP bars. Figure 6 indicates that hybrid GFRP/steel reinforced concrete beams exhibit a significant reduction in stiffness after yielding of steel in comparison with hybrid CFRP/steel reinforced concrete section. This behavior is mainly attributed to the lower elastic modulus of GFRP bars than that of CFRP bars leading to a reduced effective moment of inertia and hence larger curvature. It can be also observed from Figure 6 that increasing the tensile FRP reinforcement slightly increases the flexural stiffness of either CFRP/steel or GFRP/steel concrete members, especially after yielding of steel.
5. Failure modes of hybrid FRP/steel reinforced concrete beams For concrete beams reinforced with either steel or FRP, the balanced reinforcement ratio is attained when the beam fails by crushing of concrete in compression and either yielding of steel or rupture of FRP in tension, simultaneously. However, a failure condition of hybrid FRP/steel reinforced concrete beams, in which crushing of concrete, yielding of steel and rupture of FRP simultaneously take place, is almost impossible to occur in practice. The steel reinforcement would have yielded long before rupture of FRP
14
reinforcement as the steel yield strain (y ≈0.002) is far much less than the FRP rupture strain (f ≈0.01). Hence, the balanced condition for hybrid FRP/steel reinforced concrete beams is proposed in a way such that concrete crushing in compression and rupture of FRP reinforcement simultaneously occur, while steel reinforcement would have already yielded [12]. Figure 7 presents the strain distribution for various modes of failure as explained below in more details. 5.1 Mode I: Tensile rupture of FRP bars When εf =εfu and εc =εcu simultaneously occur, the strain distribution is unique and the neutral axis depth xl is calculated using Eq. (6) with εf and εc replaced by εfu and 0.0035, respectively:
0.0035 0.0035 fu
xl d f
(16)
When the area Af of FRP reinforcement is below a certain value Afl, FRP brittle rupture is likely to occur before concrete crushing. For a given area As of steel reinforcement, the limiting area Afl of FRP bars can be readily calculated from Eq. 12(b) with fs and ff replaced by fy and ffu, respectively:
n
Afl
f i 1
ci
bhi A'f ( s ) E 'f ( s ) 'f ( s ) - As f y (17)
f fu
where ε'f(s) is calculated from Eq. (8) with x and εc replaced by xl and 0·0035, respectively. The area Afl of FRP bars (given by Eq. (17)) forms a lower limit for the amount of FRP bars in order to avoid tensile FRP rupture. Figure 7(a) shows the strain distribution in case of Af < Afl.
15
5.2 Mode II: Yielding of steel reinforcement followed by crushing of concrete This mode of failure is characterized by yielding of steel reinforcement followed by crushing of concrete, before rupture of FRP reinforcement (see Figure 7(b) for strain distribution). In this case, the area of FRP bars is greater than Afl, therefore tensile rupture of FRP bars would not occur. This case is similar to the under-reinforced section of conventional reinforced concrete beams. The upper limit to the area of FRP bars, Afu, is reached when strains εc in the extreme compression fibre of concrete and εs in the tension reinforcement simultaneously reach εcu and εy, respectively. By following similar calculations to those presented above for mode I, the neutral axis depth xu and the area Afu of FRP bars corresponding to this upper limit of FRP bars are calculated from:
xu d s
0.0035 0.0035 y n
Afu
f i 1
ci
(18)
bhi A'f ( s ) E 'f ( s ) 'f ( s ) - As f y (19)
Ef f
where ε'f(s) and εf are calculated from Eqs. (6) and (8), respectively, with x and εc replaced by xu and 0·0035, respectively. The neutral axis depth xu given by Eq. (18) is the same as that defining the balanced section for conventional reinforced concrete beams. To ensure ductile flexural behavior, the area of FRP bars provided should be smaller than Afu given in Eq. (19) and greater than Afl to achieve ductile failure by yielding of steel reinforcement and crushing of concrete, before rupture of FRP reinforcement.
16
5.3 Mode III: Crushing of concrete before yielding of steel reinforcement (overreinforced case) If the area of FRP bars used is greater than Afu, the concrete strain εc reaches the ultimate value εcu before any yielding of tension reinforcement or FRP rupture as shown in Figure 7(c). In this case, there is a large amount of steel and FRP bars and the section is overreinforced. Such failure, often explosive, occurs with little warning, similar to that of conventional over reinforced concrete beams.
5.4 Effect of size of steel and FRP bars on the bending capacity Figure 8 presents the effect of steel and FRP reinforcements on the normalized moment capacity η (= Mf/bd2): Figure 8(a) shows the variation of the normalized moment capacity η against the steel reinforcement ratio ρs (= 100As/bd) whereas Figure 8(b) gives the variation of the normalized moment capacity η against the FRP reinforcement ratio ρf (= 100Af/bd). The dotted lines in Figure 8 represent the boundaries of different flexural failure modes as explained above. Overall, the effect of FRP reinforcement is more pronounced for hybrid FRP/steel reinforced concrete sections having less area of steel reinforcement, especially when mode II dominates the flexural behaviour of hybrid reinforced concrete beams. However, the increase in the normalized moment capacity η with the increase of hybrid FRP/steel reinforcement is insignificant when mode III (crushing of concrete before yielding of steel and rupture of FRP) dominates the flexural failure. The higher the amount of FRP bars, the less the rate of increase of the normalized moment capacity η for the same steel reinforcement ratio ρs. Therefore, it is recommended that the upper limit to the area of FRP reinforcement, Afu given by (Eq.
17
(19)), be selected as it shows optimum use of FRP reinforcement and exhibits more ductility.
6. Prediction of deflections A simplified method for deflection calculation of hybrid FRP/steel reinforced concrete beams is presented from the moment-curvature relationship developed above. The beam flexural rigidity, EIeff, corresponding to the maximum moment along the beam is firstly determined from the moment-curvature relationship at each loading as in Eq. (20) below:
EI eff
M
(20)
The midspan deflection, of hybrid FRP/steel reinforced concrete beams is then calculated using the elastic deflection formula for the loading pattern, for example, the mid-span deflection of simply supported beams loaded with a mid-span point load, P, could be calculated from Eq. (21) below:
PL3 48 EI eff
(21)
On the other hand, the immediate deflection of simply supported beams loaded with twopoint loads, each P/2, could be also computed from Eq. (22) as follows:
( P / 2)(3L2 4a 2 ) 24 EI eff
(22)
where a is the shear span and L is the span length. Bischoff [31] recently developed the following effective moment of inertia equation that showed a good agreement with test
18
results for the prediction of deflections of either steel or FRP reinforced concrete beams and also hybrid FRP/steel reinforced concrete beams up to the serviceability loading level [10].
I eff
I2 2 1 1 I 2 M cr I M 1
(23)
where Mcr is the flexural cracking moment, M is the applied bending moment, I1 (=bh3 /12) is the gross moment of inertia and I2 is the moment of inertia of transformed cracked section considering the contribution of steel and FRP reinforcement to the stiffness, Mcr (=2fcrI1 /h) is the cracking moment, fcr (= 0.62 f c ) is the modulus of rupture of concrete. The mid-span experimental deflections of six hybrid FRP/steel reinforced concrete simply supported beams tested by Lau and Pand [4] and Qu et al. [5] are compared with the predictions from the current numerical technique and Bischoff’s equation; see Figure 9(a-f). The geometrical dimensions, reinforcement details and material properties of hybrid FRP/steel reinforced concrete beams considered are given in Table 1. The deflection results obtained from the present numerical technique are in good agreement with the test results for the applied loads up to failure. Figure 9 indicates that the present technique is able to predict both the pre and post cracking deflections and to capture the beam stiffness and deflection after yielding of steel. Figure 9 also shows that the deflections predicted by Bischoff model compare well at low load levels before and after
19
the first cracking load. However, for high load levels, the predictions of Bischoff model significantly underestimate the experimental results. Fig. 10 shows the variation of the flexural rigidity, EIeff, of three hybrid beams against the applied moment. The moment-effective flexural rigidity curve consists of different segments, denoting the pre-cracking, post-cracking, yielding of steel, FRP rupture and concrete crushing stages. In the pre-cracking stage, the beams showed the highest flexural stiffness that is gradually decreased with the increase in the applied moment. Once the beams cracked, the flexural rigidity was notably reduced but remained almost constant until yielding of steel reinforcement. A further reduction of EIeff is depicted after the yielding of tensile steel reinforcement up to concrete crashing failure in case of over reinforced beams or FRP rupture in case of under reinforced beams.
7. Conclusions A numerical technique based on equilibrium of forces and full compatibility of strains has been developed to evaluate the moment-curvature relationship, moment capacity and deflections of hybrid FRP/steel reinforced concrete beams and
reinforced concrete
beams strengthened with NSM FRP bars/strips. Comparisons between experimental results available in the literature and predicted curvature, moment capacity and deflection of hybrid FRP/steel reinforced concrete beams show good agreement. The proposed model also gives good results for the moment capacity of reinforced concrete beams strengthened with NSM FRP bars/strips. The proposed technique is able to predict both the pre and post cracking deflections up to the beam failure and to capture the beam behaviour after yielding of steel or rupture of FRP reinforcements. The Bischoff model
20
for deflection prediction compares well at low load levels before and after the first cracking load, but, significantly underestimates the experimental results for high load levels. A parametric study concluded that the contribution of steel reinforcement to FRP concrete beams provides ductility and stiffness improvement of beams studied. The FRP reinforcement was found to play an important role to resist loading after yielding of steel in over reinforced sections. The results also indicated that hybrid GFRP/steel reinforced concrete beams exhibit a significant reduction in stiffness after the initiation of first crack and yielding of steel reinforcement in comparison with hybrid CFRP/steel concrete beams.
ACKNOWLEDGMENT The first author acknowledges the financial support of the Scientific and Technological Research Council of Turkey (TUBİTAK).
References [1] Tan KH, Behaviour of hybrid FRP-steel reinforced concrete beams, Proc., 3rd Int. Symp. on Non-Metallic (FRP) Reinforcement for Concrete Structures (FRPRCS-3), Japan Concrete Institute, Sapporo, 1997, 487–494. [2] Aiello MA, Ombres L, Structural performances of concrete beams with hybrid (fiberreinforced polymer-steel) reinforcements, J Compos Constr, 2002, 6(2), 133–140. [3] Toutanji HA, Saafi M. Flexural behavior of concrete beams reinforced with glass fiber-reinforced polymer (GFRP) bars. ACI Struct J 2000;97(5):712–9.
21
[4] Benmokrane B, Chaallal O, Masmoudi R. Glass fibre reinforced plastic (GFRP) rebars for concrete structures. Constr Build Mater 1995;9(6):353–364. [5] Ashour AF. Flexural and shear capacities of concrete beams reinforced with GFPR bars. Constr Build Mater 2006;20:1005–15. [6] Brown VL, Bartholomew CL. FRP reinforcing bars in reinforced concrete members. ACI Mater J 1993;90(1):34–9. [7] Pecce M, Manfredi G, Cosenza E. Experimental response and code models of GFRP RC beams in bending. J Compos Constr 2000;4(4):182–90. [8] Barris C, Torres LI, Turon A, Baena M, Catalan A. An experimental study of the flexural behaviour of GFRP RC beams and comparison with prediction models. Compos Struct 2009; 91: 286-295. [9] Ashour AF, Habeeb MN.Continuous concrete beams reinforced with CFRP bars, Struct Build 2008; SB6:349-357. [10] Wenjun Q, Zhang X, Huang H, Flexural behavior of concrete beams reinforced with hybrid (GFRP and steel) bars, J Compos Constr, 2009, 13(5), 350–359. [11] Leung HY, Balendran RV, Flexural behaviour of concrete beams internally reinforced with GFRP rods and steel rebars. Struct Surv, 2003, 21(4), 146–157. [12] Lau D, Pam HJ, Experimental study of reinforced concrete beams, Engrg Struct, 2010, 32, 3857-3865. [13] Yoon YS, Yang JM, Shin HO, Flexural strength and deflection characteristic of high-strength concrete beams having hybrid reinforcements of FRP and steel bars, In: 10th International symposium on fiber-reinforced polymer reinforcement for concrete structures 2011, FRPRC-10, in conjunction with the ACI Spring 2011
22
Convention, Farmington Hills, MI 48331, United States, American Concrete Institute; 2011. [14] Mohamed A.S. Flexural behavior and design steel-GFRP reinforced concrete beams. ACI Mater J 2013;110(6):677–85. [15] Hawileh, RA. Finite element modeling of reinforced concrete beams with a hybrid combination of steel and aramid reinforcement. Mater Des, 2015, 65, 831-839. [16] De Lorenzis L, Nanni A, and La Tegola A. Strengthening of Reinforced Concrete Structures with Near Surface Mounted FRP Rods. International Meeting on Composite Materials, PLAST 2000, Milan, Italy, 2000, 8 pp. [17] Kamal MM, Safan M A and Salama RA. Experiments on Strengthened Concrete Beams with Hybrid Steel-GFRP Reinforcement. Engrg Res J, 2008, 31(2), 201210. [18] Almusallam HT, Elsanadedy MH, Al-Salloum AY, Alsayed HS. Experimental and numerical investigation for the flexural strengthening of RC beams using nearsurface mounted steel or GFRP bars, Constr Build Mater 2013; 40: 145-161. [19] Hong KN, Han JW, Seo DW and Han SH. Flexural response of reinforced concrete members strengthend with near-surface-mounted CFRP strips. Int J Phys Scienc, 2011, 6(5): 948-961. [20] Karimi MS, Nasseri M and Maroofi E. Using NSM method in R/C beams strengthened by FRP Strips. 6th National Congress on Civil Engineering, April 2627, 2011, Semnan University, Semnan, Iran. [21] Al-Mahmoud F, Castel A, François R, Tourneur C. Strengthening of RC members with near-surface mounted CFRP rods, Compos Struct, 2009, 91, 138–147.
23
[22] De Lorenzis L, Teng JG. Near-surface mounted FRP reinforcement: an emerging technique for strengthening structures. Compos Part B 2007;38(2):119–43. [23] Soliman S, El Salakawy E, Benmokrane B. Bond performance of near-surface mounted FRP bars. J Compos Constr 2011;15(1):103–11. [24] EI-Hacha R, Rizkalla SH. Near surface-mounted fiber-reinforced polymer reinforcements for flexural strengthening of concrete structures. ACI Struct J 2004;101(5):717–26. [25] Soliman SM, El-Salakawy E, Benmokrane B. Flexural behaviour of concrete beams strengthened with near surface mounted fibre reinforced polymer bars. Can J Civil Eng 2010;37(10):1371–82. [26] Yost JR, Gross SP, Dinehart DW, Mildenberg JJ. Flexural behavior of concrete beams strengthened with near surface mounted CFRP strips. ACI Struct J 2007;104(4):430–7. [27] De Lorenzis L, Lundgren K, Rizzo A. Anchorage length of near-surface mounted fiber-reinforced polymer bars for concrete strengthening-experimental investigation and numerical modeling. ACI Struct J 2004;101(2):269–78. [28] Badawi M, Soudki K. Flexural strengthening of RC beams with prestressed NSM CFRP rods – Experimental and analytical investigation. Constr Build Mater 2009;23:3292-3300. [29] Nordin H, Taljsten B. Concrete beams strengthened with prestressed near surface mounted CFRP. J Compos Const, 2006, 10, 60-68. [30] CEB-FIP. Model code for concrete structures, Comite Euro-International du Beton, Bulletin 213/214; 1990.
24
[31] Bischoff PH, Deflection calculation of FRP reinforced concrete beams based on modifications to the existing Branson equation, J Compos Constr, 2007, 11(1), 4– 14.
25
ft
fc Parabolic curve
ftu
f c'
Ec
co
Et
c
cu
(a) Concrete in compression
ct
fs
(b) Concrete in tension ff
fy
ffu
s
Es
y
t
μct
f
Ef
fu (d) FRP composites
(c) Steel
Fig. 1 FRP, steel and concrete stress–strain relationships.
26
d'f(s)
εc
b A'f(s)
xi
h
df(s)
ε'f(s)
x Neutral axis
Cf(s) Fc
εi
segment i Concrete tensile stress
Af + As
(a) hybrid reinforced concrete section
εf and εs (b) Strain distribution
Concrete compressive stress
Tf+ Ts
(c) Stress distribution and forces
Fig. 2 Strains, stresses and forces of concrete section reinforced with FRP/steel bars
27
300 250
Mth(kNm)
200 150 100 50 0 0
50
100
150
200
250
300
Mexp(kNm)
Fig. 3 Experimental versus predicted moment capacities of hybrid FRP/steel reinforced concrete beams
28
Fig. 4 Comparison of predicted and experimental moment-curvature relationships of three hybrid FRP/steel reinforced concrete beams.
29
60
Moment(kNm)
45
30
15
FRP R/C Beam(Af=307 mm2) Steel R/C Beam(As=402 mm2) Hybrid R/C Beam (As=157 mm2, Af=307 mm2)
0 0
0.03
0.06
0.09
Curvature (1/m)
(a) Under reinforced sections
100
Moment(kNm)
80 60 40 Hybrid R/C Beam (As=402, Af=402mm2) Hybrid R/C Beam (As=509, Af=402mm2) Steel R/C Beam(As=763 mm2) FRP R/C Beam(Af=763 mm2) Hybrid R/C Beam (As=450, Af=402 mm2)
20 0 0
0.01
0.02
0.03
0.04
0.05
0.06
Curvature (1/m)
(b) Over reinforced hybrid FRP/steel sections (Af remained constant in hybrid FRP/steel sections)
30
100
Moment(kNm)
80 60 40 Hybrid R/C Beam (As=402, Af=307mm2) Hybrid R/C Beam (As=402, Af=402mm2) Steel R/C Beam (As=763 mm2) FRP R/C Beam (Af=763 mm2) Hybrid R/C Beam (As=402, Af=509mm2)
20 0 0
0.01
0.02
0.03
0.04
0.05
0.06
Curvature (1/m)
(c) Over reinforced hybrid FRPRC sections (As remained constant in hybrid FRP/steel sections) Fig. 5 (a-c) Moment-curvature relationships of under and over reinforced concrete sections
31
Fig. 6 Effect of type of FRP reinforcement (GFRP or CFRP) on the moment curvature relationships of hybrid FRP/steel reinforced concrete beam sections
32
x≤xl
ε'f(s)
xl
εs> εsy
εf= εfu (a)
εc=0.0035
εc=0.0035
εc≤ 0.0035
ε'f(s)
x>xu
εs≥ εsy
εf< εfu (b)
ε'f(s)
εs< εsy
εf< εfu (c)
Fig. 7 Strain distribution for different amount of FRP bars at failure (a) FRP rupture before concrete crushing (Af ≤ Afl) (b) Case of (Afl < Af ≤ Afu) (c) Over-reinforced case (Af > Afu)
33
10 Boundary of over-reinforced case
9 8
ρf=4 ρf=3
η=Mf /bd2
7 6
Ef=40 kN/mm2 ffu=780 N/mm2 bw=180 mm h= 280 mm df=250 mm ds=220mm
ρf=2
5
ρf=1
4
ρf=0.5
3 ρf=0.2 ρf=0.1
2 1
Boundary of GFRP rupture
ρf=0
0 0
0.5
1
1.5
2
2.5
3
3.5
4
100 A s /bd
(a) Variation of moment capacity against the area of steel reinforcement
9
Boundary of over-reinforced case
ρs=4
8 ρs=3
7 6
η=Mf /bd
2
ρs=2
5 ρs=1
4
Ef=40 kN/mm2 ffu=780 N/mm2 bw=180 mm h= 280 mm df=250 mm ds=220mm
3 2 ρs=0
Boundary of GFRP rupture
1 0 0
0.5
1
1.5
2
2.5
3
100 A f /bd
(b) Variation of moment capacity against the area of FRP bar Fig. 8 (a-b) Effect of steel and FRP reinforcements on the normalised moment capacity 160 140 (kN)
120 100
34
250
Load(kN)
200 150 100 G06T1-Numerical G06T1-Experimental Bischoff's equation
50 0 0
20
40
60
80
Midspan deflection(mm)
(b)
35
100
120
300 250 Load(kN)
200 150 100 G1T07-Numerical G1T07-Experimental Bischoff's equation
50 0 0
20
40
60
80
100
120
Midspan deflection(mm)
(c)
160
Load(kN)
120 80 40
B5-Numerical B5-Experimental B5-Bischoff's equation
0 0
5
10
15
20
Midspan deflection(mm)
(d)
36
25
30
200
120 80 B6-Numerical B6-Experimental B6-Bischoff's equation
40 0 0
5
10
15
20
25
Midspan deflection(mm)
(e)
120
Load(kN)
Load(kN)
160
80
40 B7-Numerical B7-Experimental B7-Bischoff's equation
0 0
5
10
15
20
25
30
35
40
Midspan deflection(mm)
(f) Fig. 9 (a-f) Comparison between experimental and theoretical deflections of 6 hybrid FRP/steel reinforced concrete beams tested in the literature
37
concrete cracking
steel yielding point
FRP rupture
Fig. 10 Effective flexural stiffness variations
38
Table 1. Details of hybrid steel/FRP reinforced concrete beams tested in the literature. Referen Beam ce notatio n [1]
A2
A3
B2
[2]
A1
88.3 A2
157.0 A3
C1
[10]
B3
B4
Type of loadin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g 49.0
Wid th (mm ) 175
Over all depth (mm) 350
Spa n (m m) 230 0
f c' (MP a)
Af (mm2 )
As Ef 2 (mm (kN/m ) m2)
Type of FRP
31.0
50.2
150. 7
52.0
AFR P
175
350
230 0
30.9
25.1
301. 4
52.0
AFR P
200
400
230 0
30.0
25.1
150. 7
52.0
AFR P
150
200
270 0
38.8
100. 5
270 0
38.8
100. 5
200
270 0
38.8
226. 1
235. 5
50.1
AFR P
200
270 0
38.8
100. 5
88.3
49.0
AFR P
250
180 0
28.1
226. 1
253. 2
45.0
GFR P
250
180 0
28.1
201. 0
396. 9
41.0
GFR P
AFR P 150 200
Four point bendin g 50.1 AFR P Four 150 point bendin g Four 150 point bendin g Four 180 point bendin g Four 180 point bendin g
39
B5
B6
B7
B8
[11]
L2
L5
H2
[12]
G03M D1
G1T07
G06T1
[13]
CS
GS
Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Three point bendin g Three point bendin g Three point bendin g Four point bendin g Four point
180
250
180 0
29.2
401. 9
141. 7
37.7
GFR P
180
250
180 0
29.2
401. 9
253. 2
45.0
GFR P
180
250
180 0
34.6
113. 0
141. 7
37.7
GFR P
180
250
180 0
34.6
1205 .8
396. 9
41.0
GFR P
150
200
220 0
28.5
141. 7
157. 0
40.8
GFR P
150
200
220 0
28.5
212. 5
157. 0
40.8
GFR P
150
200
220 0
48.5
141. 8
157. 0
40.8
GFR P
280
380
420 0
41.3
283. 5
981. 7
39.5
GFR P
280
380
420 0
39.8
981. 7
628. 3
38.0
GFR P
280
380
420 0
44.6
567. 1
981. 7
39.5
GFR P
230
250
190 0
75.9
128. 0
485. 0
146.2
CFR P
230
250
190 0
75.9
381. 0
485. 0
48.1
GFR P
40
[14]
B10/6S
B10/8S
B12/6S
B12/8S
[18]
RW1F
RW2F
[19]
ALII-2
AMI-2
AMII2
AMIII -1
AMIII -2
bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendin g Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng Four point bendi ng
100
200
122 0
30.0
56.6
157. 0
41.0
GFR P
100
200
122 0
30.0
100. 6
157. 0
39.0
GFR P
100
200
122 0
30.0
56.6
226. 0
41.0
GFR P
100
200
122 0
30.0
100. 6
226. 0
39.0
GFR P
150
200
200 0
36.6
78.5
157
40
GFR P
150
200
200 0
36.6
157
78.5
40
GFR P
250
400
360 0
27
56
595. 37
165
CFR P
250
400
360 0
27
56
379. 84
165
CFR P
250
400
360 0
27
56
595. 37
165
CFR P
250
400
360 0
27
28
859. 12
165
CFR P
250
400
360 0
27
56
859. 12
165
CFR P
41
AMIII -3
[20]
[21]
[28]
Four point bendi ng BMIV- Four 1 point bendi ng BMIV- Four 2 point bendi ng BMIV- Four 3 point bendi ng 6-1Fa Four point bendi ng 6-2Fa Four point bendi ng 9-1Fa Four point bendi ng 9-2Fa Four point bendi ng 12-1Fa Four point bendi ng 12-2Fa Four point bendi ng SC-12 Four point bendi ng Nonpr Four s point
250
400
360 0
27
84
859. 12
165
CFR P
400
200
360 0
27
28
859. 12
165
CFR P
400
200
360 0
27
56
859. 12
165
CFR P
400
200
360 0
27
84
859. 12
165
CFR P
152. 4
190
274 3
37.2
37.5
402
136
CFR P
152. 4
190
274 3
37.2
75
402
136
CFR P
229
190
274 3
37.2
37.5
398
136
CFR P
229
190
274 3
37.2
75
398
136
CFR P
305
190
274 3
37.2
37.5
398
136
CFR P
305
190
274 3
37.2
75
398
136
CFR P
150
280
280 0
35.1
113
226
146
CFR P
152
254
330 0
47.9
70.8 5
196. 94
136
CFR P
42
[29]
BM1
BM2
bendi ng Four point bendi ng Four point bendi ng
200
300
360 0
64
100
402
250
CFR P
200
300
360 0
65
100
402
250
CFR P
Note: Ef is the modulus of elasticity of FRP longitudinal bars, Af and As are FRP and steel reinforcement areas and f c' is the cylinder compressive strength of concrete. If the concrete cylinder compressive strength f c' is not measured in tests, it is assumed that f c' 0.85 f cu , where
f cu is the concrete cube compressive strength.
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Table 2. Comparisons between the theoretical and experimental flexural moment capacities of hybrid steel/FRP reinforced concrete beams tested in the literature.
Reference [1]
[2]
[10]
[11]
[12]
[13] [14]
[18] [19]
[20]
Beam notation A2 A3 B2 A1 A2 A3 C1 B3 B4 B5 B6 B7 B8 L2 L5 H2 G03MD1 G1T07 G06T1 CS GS B10/6S B10/8S B12/6S B12/8S RW1F RW2F ALII-2 AMI-2 AMII-2 AMIII-1 AMIII-2 AMIII-3 BMIV-1 BMIV-2 BMIV-3 6-1Fa 6-2Fa 9-1Fa
Mexp (kN) 36.00 43.50 35.25 25.14 28.41 35.55 25.14 38.38 39.66 36.36 42.57 23.55 63.30 22.23 23.07 21.11 147.00 261.00 229.00 78.20 82.80 14.09 14.41 14.89 16.33 21.84 23.03 145.9 110.15 147.62 156.36 174.08 180.44 66.77 71.09 78.69 30.37 33.06 34.13
Mthe (kN)
34.42 43.79 28.16 19.93 25.49 33.24 25.10 38.17 42.64 42.36 51.74 29.39 78.50 17.17 19.51 19.56 141.78 231.60 245.11 67.49 77.28 13.63 15.16 15.09 16.37 17.62 19.29 147.34 117.89 147.46 166.91 185.48 200.65 66.78 73.22 78.68 31.37 35.17 33.54
44
Mthe / Mexp 0.96 1.00 0.80 0.79 0.90 0.94 1.00 0.99 1.07 1.17 1.21 1.25 1.24 0.77 0.85 0.93 0.97 0.89 1.07 0.86 0.93 0.97 0.95 0.99 1.00 0.81 0.84 1.01 1.07 1.00 1.07 1.07 1.11 1.00 1.03 1.00 1.03 1.06 0.98
Experimental mode of failure SY-FRPR SY-FRPR SY-FRPR SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-FRPR SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-FRPR SY-FRPR SY-CC SY-FRPR SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC SY-CC CC CC CC*
[21] [28] [29]
9-2Fa 12-1Fa 12-2Fa SC-12 Nonprs BM1 BM2
44.37 35.83 41.08 64.4 53.08 79.3 79.3
38.69 34.78 43.75 68.8 54.32 87.3 85.1 Average Standard deviation (%) Coefficient of variation (%)
0.87 0.97 1.06 1.07 1.02 1.1 1.07 0.99 11.4%
CC CC* CC* SY-CC SY-FRPR SY-FRPR SY-FRPR
11.51%
Note: Mexp and Mthe are the experimental and theoretical moment capacities of hybrid steel/FRP sections. Experimental failure modes: SY-CC refers to steel yielding followed by concrete crushing and SY-FRPR indicates steel yielding followed by FRP rupture modes of failure, CC refers to concrete crushing. *Indicates disagreement between predicted and experimentally observed flexural failure modes.
45