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Flexural analysis of RC beam strengthened by partially de-bonded NSM FRP strip
Accepted Manuscript Flexural analysis of RC beam strengthened by partially de-bonded NSM FRP strip Soo-Yeon Seo, Moon Sung Lee, Luciano Feo PII:
S1359-8368(16)30372-9
DOI:
10.1016/j.compositesb.2016.06.056
Reference:
JCOMB 4401
To appear in:
Composites Part B
Received Date: 16 April 2016 Revised Date:
30 May 2016
Accepted Date: 11 June 2016
Please cite this article as: Seo S-Y, Lee MS, Feo L, Flexural analysis of RC beam strengthened by partially de-bonded NSM FRP strip, Composites Part B (2016), doi: 10.1016/j.compositesb.2016.06.056. 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.
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Flexural Analysis of RC Beam Strengthened by Partially De-bonded NSM FRP Strip
1
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Soo-Yeon Seo1, Moon Sung Lee2 and Luciano Feo3
Department of Architectural Engineering, Korea National University of Transportation, Chungju, Korea
2
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(Tel. 82-43-841-5205, Fax. 82-43-841-5200, email: [email protected], Corresponding author) Division of Architecture & Architectural Engineering, Hanyang University, ERICA Campus,
3
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Ansan, Korea
Department of Civil Engineering, University of Salerno, Fiscano (SA), Italy
ABSTRACT
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With regards to the Near Surface-Mounted Retrofit method using FRP in reinforced concrete flexural members, uniform deformation occurs as a whole in de-bonded FRP in the case of partial bonding at both ends and de-bonding in the central part. This leads to relief a rapid
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increase of the deformation at a particular position in the case of full bonding, which is
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effective in relieving brittle failure phenomena. In this manner, the analytical method to describe the flexural behavior of reinforced concrete beams strengthened with Near SurfaceMounted FRP strips is studied in this paper by making recourse to an analytical approach. Especially in order to improve the structural deformation capacity, the characteristics of anchorage part and de-bonded part were taken into account in the analysis conducted with respect to the beam with partially de-bonded FRP in central part. In this regard, an effective method for flexural analysis of the partially de-bonded FRP strip is proposed in the present study by reflecting the bond behavior of NSM FRP strip in the flexural model based on the
ACCEPTED MANUSCRIPT stress transfer mechanism between NSM FRP strip and concrete. From the assumption that the strain of FRP at the de-bonded region is uniform and the consideration of anchorage failure patterns, it was found that the flexural behavior of the
described to explain the ductile behavior due to the de-bonding.
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reinforced concrete beam strengthened by partially de-bonded NSM FRP strip can be suitably
Keywords: Near Surface-Mounted Retrofit; Flexural behavior of reinforced concrete beams;
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Analytical approach; Partially de-bonded FRP strip; Bond behavior of NSM FRP strip
ACCEPTED MANUSCRIPT 1. Background Retrofit of concrete member in use of FRP (Fiber Reinforced Polymer) depends on what kind of FRP as retrofit material and how to retrofit. The most general method is external
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retrofit by attaching or FRP shaped sheet or plate, to concrete surface with epoxy, especially, for horizontal member such as beam. Therefore, there have been many research works to develop the retrofit technique and find suitable analytical approach for FRP-concrete
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interface relationship [1-9].
Recently, Near Surface-Mounted Retrofit (NSMR) method has been widely developed
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and utilized in building retrofit by making recourse to fiber reinforced polymer (FRP). Due to the embedment of FRP in the concrete cover, the NSMR method not only exhibits a relatively higher strengthening efficiency compared to the external bonding method, but also has an advantage in that the reinforcement is not relatively sensitive to the external environment as it
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is positioned inside the member.
On the other hand, this method requires additional work to form grooves with over a certain size in the concrete to embed FRP reinforcement, which lowers field applicability. In
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order to make up for this shortcoming, there have been studies on the method which ensures relatively easy construction due to the embedment of vertically upright FRP strips inside the
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narrow and deep grooves formed in concrete members. The results of many previous studies have reported that this method boasts excellent strengthening efficiency due to easy groove operation and high bond force of FRP strips inserted into the groove. In addition, as a method that can evaluate the strength in the NSMR using FRP strip was presented through bond and member experiments as well as theoretical research, it has been applied as an actual practical method.
ACCEPTED MANUSCRIPT Recently, in the case of NSMR by using FRP, a series of studies have been conducted on how to increase the deformation capacity of flexural members and improve the construction ability by de-bonding FRP in the central part based on the consideration of its high bond
improving the flexural deformation capacity.
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capacity. In this case, the uniform strain of FRP in the de-bonded region can contribute to
In this regard, this study seeks to proceed with the analytical research to describe the
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flexural behavior of the Near Surface-Mounted (NSM) FRP strip de-bonded in the central part. By analyzing a series of research results related to the bond behavior of NSM FRP strip
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and utilizing the results, it attempts to conduct a flexural analysis that reflects the bond properties of NSM FRP on the tension-side. Some researchers [10, 11] proposed an analytical method that can describe the flexural behavior of the members strengthened by partially debonded FRP based on the curvature distribution. However, in their studies, the anchorage
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behavior that occurs in the bonded part of the strengthened FRP strip in flexure was not adequately considered, but only the material properties of FRP were taken into account. Therefore, the bond characteristics of FRP strips by the de-bonding cannot be adequately
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taken into consideration.
In this study, a flexural analysis method of RC beam strengthened by the partially de-
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bonded NSM FRP strip is proposed by describing the stress transfer mechanism between NSM FRP strip and concrete.
2. Previous research
In flexural analyses for which FRP is partially de-bonded, it is very important to properly consider the bond behavior in the bonded region. In recent years, Seo et al. [12, 13] have confirmed that the NSMR significantly improves the strength compared with externally
ACCEPTED MANUSCRIPT bonded retrofit (EBR) based on the results of experiments and theoretical studies. Also they found that the bond strength is improved corresponding to the increase of bond length and number of FRP strip lines due to the enlarged surface area to be bonded to concrete. In
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addition, they presented a correlation equation that considers the proper spacing of each FRP depending on the bond length as well as group effect and confirmed the suitability of the equation through an experiment.
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The stress transfer from NSM FRP strip to concrete is very effective since the bond shear stress of a relatively wide FRP strip surface is transferred to the concrete though epoxy.
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Theoretical contributions involving shear stress fields in composite beams and plates have been recently carried out by Barretta et al. [14-18]. The stress transfer is basically based on the theory of bond relationship between rebar and concrete. However, since the bond areas and rigidity of NSM FRP strip are large, consistent research efforts are required with respect
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to the stress transfer in the bonded interface. Seracino et al. [19] described the bond strength of NSM strip-to-concrete joints, and Seo [20] evaluated the suitability of this equation through a comparison with bond test result.
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In relation to the flexural retrofit by partially de-bonding FRP, Lees and Burgoyne [21, 22] conducted experiments and analytical studies on the beam strengthened by partially de-
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bonded aramid FRP and reported that the ductility improvement arises in the partial bebonding. Ali and Khaled [23] conducted a study on the partially bonded FRP and reported that the beam partially strengthened in the anchorage at ends ensures a higher strengthening effect and ductility compared to the fully bonded beam. Han et al. [10, 11] proposed a flexural analysis model of RC beam strengthened by the partially de-bonded FRP and demonstrated that the proposed model properly simulates the flexural behavior through an experiment. In particular, they confirmed that the ductility can be improved by the partial debonding. Their analysis method takes into account a change in the curvature distribution
ACCEPTED MANUSCRIPT according to the de-bonding and is based on the fact that FRP exerts sufficient anchorage performance at both ends of the beam. However, in this model, the anchorage behavior that occurs in the bonded part of the FRP strip was not adequately considered, but only the
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material properties of FRP were taken into account. Therefore, the bond characteristics of FRP strips by the de-bonding cannot be adequately taken into consideration.
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3. Flexural Behavior Model of RC beam strengthened by partially de-bonded NSM FRP
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3.1 Assumed condition
With regards to the NSMR using FRP in reinforced concrete flexural members, uniform deformation occurs as a whole in de-bonded FRP in the case of partial bonding at both ends and de-bonding in the central part. This leads to relief of a rapid increase of the deformation
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at a particular position in the case of full bonding, which is effective in relieving brittle failure phenomena. Fig. 1 is a picture showing the results of an experiment conducted by Seo et al. [24], and as shown in the figure, the failure is concentrated in the central part of the
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beam in the case of the full bonding, whereas cracks are distributed at appropriate intervals in the case of the de-bonding at the central part. It means that a slightly ductile behavior can be
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induced with an increase in the uniform deformation of FRP. However, in order to show such a behavior, FRP should extend a sufficient length to be anchored at ends. In order to represent the flexural behavior of the beam strengthened by partially debonded FRP, the following assumptions that are commonly defined in the reinforced concrete beams can be set forth as preconditions 1) Strain distribution of the section in the bonding area is linear. 2) There is no friction between concrete and FRP in the de-bonded region. 3) Moment is assumed to be linearly distributed with respect to the length.
ACCEPTED MANUSCRIPT 4) For concrete, a parabolic stress-strain relationship proposed by Collins and Mitchell [25] is used, and the concrete is assumed to be continuously deformed even after reaching strain at failure (Fig.2 (a)). 2
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ε ε fc f ' = 2 ⋅ c − c , ε c ' = 2 ⋅ c , Ec = 8500 ⋅ 3 f c ' fc ' εc' εc' Ec
(1)
5) Stress-strain relationship in the tension of concrete is linear and ignored after cracks
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occur. 6) Rebar is assumed to be a bi-linear model (Fig.2 (b)).
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7) Stress-strain relationship of the partially anchored FRP increases linearly up to the bond strength, and is assumed to be deformed until it reaches the maximum strain of FRP after reaching the bond strength (Fig. 2(c)).
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3.2 Bond strength of NSM FRP
As shown in Fig.3, if tensile force is applied to the FRP strip embedded in concrete, the failure modes according to the bond length can be divided into tensile failure strength of FRP
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( T ff ), epoxy bond failure strength ( T ft ) and bond failure strength ( T fb ). Since the tensile force of FRP in the flexural member is determined by the weakest among these three
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mechanisms, it can be summarized as follows.
Tf = min.{Tff , Tfc , Tfb } ·
(2)
T ff = φA f f fu
(3)
T fc = λτ ef (2b f Lb )
(4)
T fb = α p 0.85 n gηφ f
0.25
f c '0.33 ( L per γ )( E f A f )
(5)
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L γ = b L per
0.65
(6)
L γ + Be ≤ 1 .0 η = per nL γ per
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(7)
L per = 2b f + t f
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φ f = bf / t f
(8)
(9)
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Eq.(5) suggested by Seo [20] based on empirical formula of Sericiano et al. [19] considers the bond length effect and the group effect arising out of a compact space between FRP strips as shown in Fig.4. The resisting forces at FRP member anchorage according to bond length, based on Eqs.(2)-(9), are shown in Table 2. It was found that the bond failure strength ( T fb ) is
region.
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governing the failure for all case except CP1600-1 specimen fully bonded without de-bonded
3.3 Behavior analysis of RC beam strengthened in flexure by partially de-bonded NSM
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FRP strip
The flexural behavior of RC beam is normally explained by its curvature distribution at a level of load. Fig.5 represents the curvature distribution depending on de-bonding in the beam strengthened by the partial de-bonding. Until the central part reaches yield moment, the moment distribution shows a tendency to increase linearly from point ‘a’ to point ‘c’. In the case of fully bonded FRP, the curvature distribution shape is similar to that of moment. However, the curvature of RC beam with de-bonded region of FRP shows a drop at point ‘b’ where de-bonding begins since the strain of FRP is uniform in the de-bonded region. If the
ACCEPTED MANUSCRIPT moment in the central part is greater than the yield moment, the yield occurs within the debonded region. Therefore, the curvature distribution is divided into three sections such as elastic part in the bonded region, elastic part in the de-bonded region and inelastic part in the
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de-bonded region. Fig.6 represents the strain distribution of concrete on the compression side and FRP on the tension side after the central moment reaches the yield.
With an increase in the flexural moment, the curvature distribution for each step is
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calculated as in Fig.5, and the deflection at each position can be estimated. Fig.7 shows the strain distributions at the position ‘c’ in which the maximum stress occurs when the member
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reaches the yield moment and maximum moment, and ‘b’ section where de-bonding begins, respectively. Since point ‘b’ is the place where bonding ends and de-bonding begins, two kinds of deformation conditions are presented in the case of FRP. In other words, since the strain of FRP is the same in the de-bonded region, FRP strain of point ‘c’ becomes the same
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as that of point ‘b’, and therefore the strain of FRP cannot form a linear relationship with those of concrete and rebar at point ‘b’, but has a relatively higher value. In addition, an equilibrium relationship where the amount of deformation of de-bonded FRP is relatively less
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compared to that of tensile rebar even at point c where the maximum moment occurs should be considered to calculate the sectional forces.
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Fig.8 shows the procedures for calculating the flexural strength and deformation of the member corresponding to the increase of moment. The calculation procedures are as follows: i) on the base of the flexural theories, calculate the moment and curvature at point ‘b’ where de-bonding begins which corresponds to the moment of point ‘a’, the central part, and the moment of point ‘c’ if point ‘a’ exceeds the yield moment, ii) find the curvature at point ‘b’ by using the equilibrium relationship of the force since FRP strain within de-bonded region such as point ‘a’ and point ‘b’ is the same, iii) with the use of the above relationship, produce
ACCEPTED MANUSCRIPT the deflection by using the degrees of curves when the moment at point ‘a’ reaches the crack moment, yield moment and maximum moment, respectively.
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Step 1. Calculation of yield moment and curvature at perfect bond condition
The strengths of concrete, rebar and FRP at perfect bond condition can be calculated as
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follows:
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C c = 0 .5 f c ⋅ c y ⋅ b
C s = Asc ⋅ f sc Ts = As ⋅ f s
(11)
(12)
(13)
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Tf = Af ⋅ f f
(10)
In consideration of the equilibrium of forces in section, the neutral axis depth c y
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satisfying the following equation can be found.
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ΣPx = Cc + Cc − Ts − Tf = 0
(14)
With the use of c y , the yield moment and yield curvature are calculated as follows.
2 M y = Cc ⋅ (c y ) + Cs ⋅ (c y − d1 ) + Ts ⋅ (d − c y ) + T f ⋅ (d f − c y ) 3 when
ε s = ε sy ,
(15)
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(d − c y )
ε sy , ε sc =
(d − c y )
ε sy , ε f =
(d f − c y ) (d − c y )
ε sy
(16)
ε sy
(17)
M yb
M yb =
and
0.5L − Ldb ⋅My 0.5L
when the center reaches
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0.5L − Ldb ⋅ φy 0.5L
φyb
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(d − c y )
Step 2. Calculation of
φ y ,b =
(c y − d 1 )
My
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φy =
cy
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εc =
Step 3. Calculation of curvature φ y ,b − d at transit section when the center reaches
(18)
(19)
My
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FRP strain in de-bonded region is all the same, and it becomes the average value of the strains at section ‘b’ and ‘c’.
ε f ,b + ε f , c
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ε f ,b − d =
2
(20)
After calculating the stress and T f corresponding to the given FRP strain, the curvature at transit section is calculated by finding the compression fiber strain of concrete to match moment at the section and obtaining the neutral axis length c y which satisfies the equilibrium relationship of the section.
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φ y ,b − d =
ε cy ,b −d c y ,b − d
< φy
(21)
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Step 4. Calculation of the ultimate moment and curvature at perfect bond condition
The strength of concrete at perfect bond condition can be calculated with equivalent
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stress block as follows.
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C c = α 1 ⋅ β 1 ⋅ f c ⋅ cu ⋅ b when ε c = ε cu ,
ε cu cu
(cu − d 1 ) , ε s =
or when ε f =
ε fu ( d f − cu )
cu
cu , ε sc =
( d − cu ) , ε f =
ε fu
( d f − cu )
ε cu cu
( d f − cu )
( cu − d 1 ) , ε f =
ε fu ( d f − cu )
(23)
( d − cu )
(24)
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εc =
ε fu ,
ε cu
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ε sc =
(22)
The stress-strain relationship of FRP reflects the anchorage condition of FRP. The maximum moment and curvature are calculated by using cu that satisfies the equilibrium condition in section after finding it through iterative calculations.
M u = Cc ⋅
cu + C s ⋅ (cu − d1 ) + Ts ⋅ (d − cu ) + T f ⋅ (d f − cu ) 2
(25)
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φu =
εc
(26)
cu
In relation to the maximum moment estimation, although the strength of member was
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calculated at perfect bond condition, the anchorage condition of the end was taken into account, and therefore the strength of the case where the anchorage is not sufficient in partial de-bonding can be properly calculated.
0.5L − Ldb ⋅ φu 0.5L
(27)
(28)
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φu ,b =
0.5L − Ldb ⋅ Mu 0.5L
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M u ,b =
φu ,b when the center reaches M u
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Step 5. Calculation of M u ,b and
Step 6. Finding the location of yielding section when the center reaches M u
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M Luy = 0.5 L1 − y Mu
Step 7. Calculation of curvature at transit section,
(29)
φu ,b−d when the center reaches M u
FRP strain in de-bonded region is all the same, and it becomes the average value of the strains at section ‘b’ and ‘c’. The value is to be its ultimate strain ε fu when the FRP tension failure governs.
ACCEPTED MANUSCRIPT After calculating the stress and T f corresponding to the given FRP strain and obtaining the neutral axis depth cu which satisfies the equilibrium relationship of the section by using
ε cu ,b − d c u ,b − d
< φu
(30)
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φ u ,b − d =
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ε c ,b = ε c ,b −d , the curvature at transit section is calculated.
the relationship of
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3.4 Analysis result and discussion
According to the aforementioned analysis procedures, calculations were performed with specimens of Seo et al. [24], and the adequacy of the analysis method was compared. The details of the specimens are shown in Fig.9 and Table 1. Table 2 shows the resisting force of FRP anchorage part depending on the bond lengths calculated using Eq.(2) ~ (10). Fig.10
experimental results.
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represents a comparison of load-displacement curve between analysis results and
The analysis results of the case where the entire lower part is strengthened by NSM FRP
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strip without de-bonded region showed that concrete on the compression side reached the
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maximum strength before FRP reached the maximum strain. In the experiment, the strength increased almost linearly until the concrete on the compression side reached the maximum strength, forming a high yield point. Moreover even after concrete on the compression side reached the maximum strength, it exerted a predetermined capacity with no rapid decrease in the internal force until FRP on the tension side suffered a fracture. This is because as a plate for loading with width of 50mm confined concrete on the compression side to some extent in the place where the compressive stress is concentrated, concrete was sufficiently resistant to compression without reaching brittle failure rapidly (refer to Fig. 11 failure mode).
ACCEPTED MANUSCRIPT Accordingly, the analysis assumed that concrete on the compression side maintains the maximum strength until FRP reaches its ultimate strain even after reaching the maximum strength, and therefore the maximum moment and curvature at fracture of FRP on the tension
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side were represented together. In the case of specimens with de-bonded region of FRP in central part, the maximum internal forces were found to be determined by the fracture of FRP anchorage part on the
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tension side. Although the stiffness until the yield was slightly high, and the amount of deflection until the strengthening effect is lost was rather low in the analysis results, the
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analysis results are in good correspondence with experimental results as a whole. Fig.12 shows the curvature distribution when the central part of member reaches the maximum strength and yield, and Fig.13 shows the strain of the compressive edge, compressive rebar, tensile rebar, and FRP at each position when the central section reaches the maximum
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moment. Since the strain distribution is linear in the bonding area, but FRP strain is uniform in the de-bonded region, the strain of FRP is relatively larger than that of tensile rebar at point ‘b’ but smaller at point ‘c’.
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With respect to the maximum internal force and yield of the member, a comparison between experimental results and analysis results is shown in Table 3. The stiffness until the
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yield was slightly high, and the amount of deflection until the strengthening effect is lost was rather low in the analysis results. Fig.14 represents the changes in load-displacement relationship due to an increase in de-bonded length, and as the de-bonded length increases, the deformation capacity is improved after yielding.
4. Conclusion
ACCEPTED MANUSCRIPT In this paper, a simple analytical method describing the flexural behavior of reinforced concrete beams strengthened with partially de-bonded NSM FRP strips has been studied. Especially, the characteristics of anchorage part and de-bonded region were reflected in the
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analysis. In this regard, in this study, it has proposed a method for flexural analysis of the partially de-bonded FRP strip by reflecting the bond behavior of NSM FRP strip in the flexural model based on the stress transfer mechanism between NSM FRP strip and concrete.
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Some concluding remarks are listed as follows.
The strain of FRP at the de-bonded region was assumed to be uniform, and the
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equilibrium of the section was satisfied with respect to the applied load. Therefore, it was found that the flexural behavior of the reinforced concrete beam strengthened by partially debonded NSM FRP strip can be described.
According to the length of the de-bonded region, failure mechanisms can be divided into
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tensile failure of FRP, tensile failure of concrete, and bond failure of epoxy, of which the dominant failure mechanism were taken into consideration to conduct flexural analysis of the case being dominated by the behavior of FRP.
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The analysis results of the case where the entire lower part is strengthened by NSM FRP strip without de-bonded region revealed that concrete on the compression side showed brittle
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failure mode after reaching the maximum strength. However, in the actual experiment, concrete on the compression side exerted a predetermined capacity with no rapid degradation of the internal force even after reaching the maximum strength, and thus there is a need to reflect this characteristic in the actual analysis.
Acknowledgement This research was supported by a grant from the Academic Research Program of Korea
ACCEPTED MANUSCRIPT National University of Transportation in 2013.
Notation
bf ⋅ t f
(mm2)
= cross-sectional area of FRP reinforcement =
As , Asc
= cross-sectional area of tension bar and compression bar, respectively
Be
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(mm2)
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Af
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= distance between centroids of FRP strips located at extreme sides in considering group effect
= width of beam and NSM FRP strip, respectively (mm)
Cc , C s
= resultant compression force of concrete and steel reinforcement,
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b , bf
respectively (N)
= neutral axis depth at yield and ultimate section, respectively (mm)
c y ,b , cu ,b
= neutral axis depth of bonded part at transit section when the beam
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c y , cu
researches yield and ultimate state, respectively (mm)
c y ,b−d , cu ,b −d
= neutral axis depth of de-bonded part at transit section when the beam researches yield and ultimate state, respectively (mm)
d , d1 , d f
= depth of tension steel reinforcement, compression steel reinforcement and FRP reinforcement, respectively (mm)
ACCEPTED MANUSCRIPT Ec , E f
= modulus of elasticity of concrete and FRP reinforcement, respectively (Mpa)
f c , fc '
= compressive stress and ultimate compressive strength of concrete,
f f , f fu
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respectively (Mpa)
= tensile stress and ultimate tensile strength of FRP reinforcement,
= stress of steel reinforcements at compression and tension, respectively (Mpa)
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f sc , f s
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respectively (Mpa)
= beam span (mm)
Lb , L db
= bonded length and de-bonded length of FRP reinforcement, respectively (mm)
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L
= circumferential length of FRP (mm)
Luy
= length from loading point to yielding section (mm)
M y ,M u
= yield and ultimate moment of beam, respectively (N.mm)
Mb
= moment at transit section (section ‘b’) (N.mm)
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L per
M y ,b , M u , b
= moment at transit section when the center reaches yield and ultimate moment, respectively (N.mm)
ng
= number of FRP strip in considering group effect
ACCEPTED MANUSCRIPT tf
= thickness of FRP strip (mm)
Ts , Tf
= resultant tension force of steel reinforcement and FRP reinforcement,
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respectively (N) = ratio of average stress in rectangular compression block
αp
= test coefficient (0.7 is used for the calculation of flexural strength of beam
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α1
in this study).
= ratio of depth of rectangular compression block to the neutral axis
εc ,εc '
= concrete strain at certain state and ultimate strength, respectively
ε c ,b , ε c ,b − d
= concrete strain of bonded part and de-bonded part at transit section,
ε cy , ε cu
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respectively
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β1
= strain at yield and ultimate state of beam, respectively
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ε cy ,b−d , ε cu ,b−d = concrete strain at transit section when the center reaches yield and ultimate
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state, respectively
ε f , ε fu
= strain of FRP reinforcement at certain level and ultimate strength, respectively
ε f , b , ε f ,b − d
= strain of FRP reinforcement of bonded part and de-bonded part at transit section, respectively
ε fy,b , ε fu ,b
= strain of FRP reinforcement of bonded part at transit section when the
ACCEPTED MANUSCRIPT center reaches yield and ultimate state, respectively
ε fy ,b−d , ε fu ,b−d
= strain of FRP reinforcement of de-bonded part at transit section when the center reaches yield and ultimate state, respectively = strain of FRP reinforcement at section ‘c’
ε fy
= strain of FRP reinforcement at yield state
ε s , ε sc
= strain of steel reinforcement in tension part and in compression part,
ε sy,b−d , ε su ,b−d
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respectively
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ε f ,c
= strain of steel reinforcement in tension part when the center reaches yield and ultimate state, respectively
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ε scy ,b−d , ε scu ,b−d = strain of steel reinforcement in compression part when the center reaches yield and ultimate state, respectively = yield strain of steel reinforcement
φ
= strength reduction factor (1.0 is used in this study) (rad/mm)
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φ y , φu
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ε sy
φu ,b , φu ,b−d
= curvature of beam at yield and ultimate moment, respectively (rad/mm)
= curvature of beam at the bonded and de-bonded part of transit section when the center reaches ultimate state, respectively (rad/mm)
φ y , b φ y ,b − d
= curvature of beam at the bonded and de-bonded part of transit section when the center reaches yield state, respectively (rad/mm)
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λ
= effective bond loss factor (1.0 is used in this study)
τ ef ,τ f
= shear strength of epoxy and bond strength of NSM FRP strip, respectively
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(Mpa)
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[4] J Yao, JG Tang, JF Chen. Experimental study on FRP-to-concrete bonded joints. Compos
[5] D’Ambrisi A., Feo L., Focacci F. Bond-Slip Relations for PBO-FRCM Materials Externally Bonded to Concrete. Compos Part B Eng 2012; 43: 2938- 2949
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[6] Ascione L., Berardi V., FEO L., Mancusi G. A numerical evaluation of the inter laminar stress state in externally FRP plated RC beams. Compos Part B Eng 2005;36: 83-90. [7] D'Ambrisi A., Feo L., Focacci F.
Experimental and analytical investigation on bond
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between Carbon-FRCM materials and masonry. Compos Part B Eng 2013:46:15- 20. [8] Yuan H., Lu X., Hui D., Feo L. Studies on FRP-concrete interface with hardening and
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softening bond-slip law. Compos Struct 2012;94:3781- 3792. [9] Aprile A., Fe L., Concrete cover rip-off of R/C beams strengthened with FRP composites. Compos Part B Eng 2007; 38:759-771. [10] Han TC, Jeffrey SW, Khaled AS. Partially bonded near-surface-mounted CFRP bars for strengthened concrete T-beams. Construction and Building Materials 2011;25:2441-2449. [11] Han TC, Jeffrey SW, Khaled AS. Analysis of the Flexural Behavior of Partially Bonded FRP Strengthened Concrete Beams, Journal of Composites for Construction 2008;12(4):375-386.
ACCEPTED MANUSCRIPT [12] Seo S, Yoon S, Kwon Y, Choi K. Bond behavior between near surface-mounted fiber reinforced polymer plates and concrete in structural strengthening. Journal of Korea Concrete Institute 2011;23(5):675-82 (written in Korean).
concrete structures, Composite Structures 2013;95:719–727.
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[13] Seo S, Feo L, Hui D. Bond strength of near surface-mounted FRP plate for retrofit of
[14] Barretta R., Luciano R., Willis J.R. On torsion of random composite beams. Composite
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Structures 2015;132: 915-922.
[15] Barretta R., Feo L., Luciano R. Some closed-form solutions of functionally graded beams
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undergoing nonuniform torsion. Composite Structures 2015;123:132-136. [16] Barretta R., Luciano R. Analogies between Kirchhoff plates and functionally graded Saint-Venant beams under torsion. Continuum Mechanics and Thermodynamics 2015;27:499-505.
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[17] Barretta R., Feo L., Luciano R. Torsion of functionally graded nonlocal viscoelastic circular nanobeams. Composites: Part B 2015;72:217-222. [18] Apuzzo A., Barretta R., Luciano R., Some analytical solutions of functionally graded
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Kirchhoff plates. Composites: Part B 2015;68:266-269. [19] Seracino R, Jones N, Ali M, Page M,. Oehlers D. Bond strength of near-surface mounted
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FRP strip-to-concrete joints. Journal of Composite Construction.2007;11(4):401–409. [20] Seo S. Bond Strength of Near Surface-Mounted FRP Strip in RC Member. Journal of the Korea Concrete Institute 2012; 24(4):415-422 (written in Korean). [21] Lees JM, Burgoyne CJ. Experimental study of influence of bond on flexural behavior of concrete beams pretensioned with aramid fiber reinforced plastics. ACI Struct J 1999;96(3):377–85. [22] Lees JM, Burgoyne CJ. Analysis of concrete beams with partially bonded composite
ACCEPTED MANUSCRIPT reinforcement. ACI Struct J 2000;97(2):252–60. [23] Ali C, Khaled S. Flexural Response of Reinforced Concrete Beams Strengthened with End-Anchored Partially Bonded Carbon Fiber-Reinforced Polymer Strips, Journal of
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Composites for Construction 2005: 9(2):170-177. [24] Seo S, Choi K, Kwon Y, Lee K. Flexural Strength of RC Beam Strengthened by Partially
and Materials 2016; Online published.
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De-bonded Near Surface-Mounted FRP Strip, International Journal of Concrete Structures
[25] Collins MP, Mitchell D. Prestressed concrete basics. Canadian Prestressed Concrete
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Institute (CPCI), Ottawa, Ontario; 1987.
Table 1 Specimen list
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Specimen name
Retrofit method
Bonded length (mm)
De-bonded length (mm)
BC2000
None
-
-
1,600
-
500 at each ends
600
300 at each ends
1000
CP1600-1
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3.6×16
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CP300-1
NSMR
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CP500-1
Strip dimension (width×height) (mm×mm)
ACCEPTED Table 2 Internal tensile forces of NSM FRP strip MANUSCRIPT Forces corresponding to the bonded length (kN)
Forces
Lb=400mm
Lb=500mm
Lb=800mm
T ff
161.28
161.28
161.28
161.28
T ft
172.80
230.40
288.00
460.8
T fb
85.44
93.81
100.87
235.02
Tf
78.07
90.12
103.62
161.28
T f / T ff
0.53
0.58
0.63
1.00
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Lb=300mm
Table 3 Comparison with test and analysis results
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Specimen
BC2000
Yield
Ultimate
Test or Analysis
Py (kN)
δ y (mm)
Ke
Pu (kN)
δ u (mm)
Test
143.18
6.03
23.74
190.24
79.61
Analysis
142.5
4.38
32.53
159.03
19.27
Test
200.55
6.72
29.84
225.55
15.98
Analysis
150.00
4.11
36.50
215.96
10.78
Test
161.32
6.35
25.40
208.39
19.06
Analysis
150.00
3.94
38.07
205.94
15.73
Test
154.95
6.63
23.37
194.17
19.16
Analysis
150.00
3.93
38.17
198.04
17.08
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CP300-1
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CP500-1
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CP1600-1
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Fracture is concentrated on the beam center
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(a) Fully bonded NSM FRP along the beam length
(b) Partially de-bonded NSM FRP around center of beam
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Fig. 1 Spreading out of cracks by fully bonded and partially de-bonded NSM FRP
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(a) Concrete
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(b) Reinforcement
(c) Bond stress-strain relationship of NSM FRP embedded in concrete Fig. 2 Stress-strain curves of materials
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Fig. 3 Internal force of partially de-bonded NSM FRP
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Fig. 4 Effective concrete width of NSM FRP strips
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Fig. 5 Moment and curvature distribution of RC beam with partially de-bonded NSM FRP
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Fig. 6 Strain distribution of partially de-bonded NSM FRP in strengthened beam
(a) Yield condition
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(b) Ultimate condition
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Fig. 7 Strain distributions of a section at yield and ultimate condition
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Fig. 8 Calculation process of the deflection of RC beam with partially de-bonded NSM FRP
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Fig. 9 Detail of the specimen done by Seo et al.
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(b) Fully bonded
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(a) RC beam w/o NSM FRP
(c) Lb = 500mm
(d) Lb =300mm
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Fig. 10 Comparison of calculation and test results in load-displacement
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Fig. 11 Failure shape of CP1600-1 specimen
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(a) Fully bonded
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(b) Lb=500mm
(c) Lb = 300mm
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Fig. 12 Curvature distribution at yield and ultimate state
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(b) at transit section (point b), de-bonded region
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(a) at transit section (point b), bonded region
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(c) at yield section
(d) at center section (point c)
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Fig. 13 Distributions of FRP strains at ultimate condition (Lb=300mm)
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Fig. 14 Load-displacement variation corresponding to the de-bonded length
S1359-8368(16)30372-9
DOI:
10.1016/j.compositesb.2016.06.056
Reference:
JCOMB 4401
To appear in:
Composites Part B
Received Date: 16 April 2016 Revised Date:
30 May 2016
Accepted Date: 11 June 2016
Please cite this article as: Seo S-Y, Lee MS, Feo L, Flexural analysis of RC beam strengthened by partially de-bonded NSM FRP strip, Composites Part B (2016), doi: 10.1016/j.compositesb.2016.06.056. 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.
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Flexural Analysis of RC Beam Strengthened by Partially De-bonded NSM FRP Strip
1
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Soo-Yeon Seo1, Moon Sung Lee2 and Luciano Feo3
Department of Architectural Engineering, Korea National University of Transportation, Chungju, Korea
2
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(Tel. 82-43-841-5205, Fax. 82-43-841-5200, email: [email protected], Corresponding author) Division of Architecture & Architectural Engineering, Hanyang University, ERICA Campus,
3
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Ansan, Korea
Department of Civil Engineering, University of Salerno, Fiscano (SA), Italy
ABSTRACT
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With regards to the Near Surface-Mounted Retrofit method using FRP in reinforced concrete flexural members, uniform deformation occurs as a whole in de-bonded FRP in the case of partial bonding at both ends and de-bonding in the central part. This leads to relief a rapid
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increase of the deformation at a particular position in the case of full bonding, which is
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effective in relieving brittle failure phenomena. In this manner, the analytical method to describe the flexural behavior of reinforced concrete beams strengthened with Near SurfaceMounted FRP strips is studied in this paper by making recourse to an analytical approach. Especially in order to improve the structural deformation capacity, the characteristics of anchorage part and de-bonded part were taken into account in the analysis conducted with respect to the beam with partially de-bonded FRP in central part. In this regard, an effective method for flexural analysis of the partially de-bonded FRP strip is proposed in the present study by reflecting the bond behavior of NSM FRP strip in the flexural model based on the
ACCEPTED MANUSCRIPT stress transfer mechanism between NSM FRP strip and concrete. From the assumption that the strain of FRP at the de-bonded region is uniform and the consideration of anchorage failure patterns, it was found that the flexural behavior of the
described to explain the ductile behavior due to the de-bonding.
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reinforced concrete beam strengthened by partially de-bonded NSM FRP strip can be suitably
Keywords: Near Surface-Mounted Retrofit; Flexural behavior of reinforced concrete beams;
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Analytical approach; Partially de-bonded FRP strip; Bond behavior of NSM FRP strip
ACCEPTED MANUSCRIPT 1. Background Retrofit of concrete member in use of FRP (Fiber Reinforced Polymer) depends on what kind of FRP as retrofit material and how to retrofit. The most general method is external
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retrofit by attaching or FRP shaped sheet or plate, to concrete surface with epoxy, especially, for horizontal member such as beam. Therefore, there have been many research works to develop the retrofit technique and find suitable analytical approach for FRP-concrete
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interface relationship [1-9].
Recently, Near Surface-Mounted Retrofit (NSMR) method has been widely developed
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and utilized in building retrofit by making recourse to fiber reinforced polymer (FRP). Due to the embedment of FRP in the concrete cover, the NSMR method not only exhibits a relatively higher strengthening efficiency compared to the external bonding method, but also has an advantage in that the reinforcement is not relatively sensitive to the external environment as it
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is positioned inside the member.
On the other hand, this method requires additional work to form grooves with over a certain size in the concrete to embed FRP reinforcement, which lowers field applicability. In
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order to make up for this shortcoming, there have been studies on the method which ensures relatively easy construction due to the embedment of vertically upright FRP strips inside the
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narrow and deep grooves formed in concrete members. The results of many previous studies have reported that this method boasts excellent strengthening efficiency due to easy groove operation and high bond force of FRP strips inserted into the groove. In addition, as a method that can evaluate the strength in the NSMR using FRP strip was presented through bond and member experiments as well as theoretical research, it has been applied as an actual practical method.
ACCEPTED MANUSCRIPT Recently, in the case of NSMR by using FRP, a series of studies have been conducted on how to increase the deformation capacity of flexural members and improve the construction ability by de-bonding FRP in the central part based on the consideration of its high bond
improving the flexural deformation capacity.
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capacity. In this case, the uniform strain of FRP in the de-bonded region can contribute to
In this regard, this study seeks to proceed with the analytical research to describe the
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flexural behavior of the Near Surface-Mounted (NSM) FRP strip de-bonded in the central part. By analyzing a series of research results related to the bond behavior of NSM FRP strip
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and utilizing the results, it attempts to conduct a flexural analysis that reflects the bond properties of NSM FRP on the tension-side. Some researchers [10, 11] proposed an analytical method that can describe the flexural behavior of the members strengthened by partially debonded FRP based on the curvature distribution. However, in their studies, the anchorage
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behavior that occurs in the bonded part of the strengthened FRP strip in flexure was not adequately considered, but only the material properties of FRP were taken into account. Therefore, the bond characteristics of FRP strips by the de-bonding cannot be adequately
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taken into consideration.
In this study, a flexural analysis method of RC beam strengthened by the partially de-
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bonded NSM FRP strip is proposed by describing the stress transfer mechanism between NSM FRP strip and concrete.
2. Previous research
In flexural analyses for which FRP is partially de-bonded, it is very important to properly consider the bond behavior in the bonded region. In recent years, Seo et al. [12, 13] have confirmed that the NSMR significantly improves the strength compared with externally
ACCEPTED MANUSCRIPT bonded retrofit (EBR) based on the results of experiments and theoretical studies. Also they found that the bond strength is improved corresponding to the increase of bond length and number of FRP strip lines due to the enlarged surface area to be bonded to concrete. In
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addition, they presented a correlation equation that considers the proper spacing of each FRP depending on the bond length as well as group effect and confirmed the suitability of the equation through an experiment.
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The stress transfer from NSM FRP strip to concrete is very effective since the bond shear stress of a relatively wide FRP strip surface is transferred to the concrete though epoxy.
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Theoretical contributions involving shear stress fields in composite beams and plates have been recently carried out by Barretta et al. [14-18]. The stress transfer is basically based on the theory of bond relationship between rebar and concrete. However, since the bond areas and rigidity of NSM FRP strip are large, consistent research efforts are required with respect
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to the stress transfer in the bonded interface. Seracino et al. [19] described the bond strength of NSM strip-to-concrete joints, and Seo [20] evaluated the suitability of this equation through a comparison with bond test result.
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In relation to the flexural retrofit by partially de-bonding FRP, Lees and Burgoyne [21, 22] conducted experiments and analytical studies on the beam strengthened by partially de-
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bonded aramid FRP and reported that the ductility improvement arises in the partial bebonding. Ali and Khaled [23] conducted a study on the partially bonded FRP and reported that the beam partially strengthened in the anchorage at ends ensures a higher strengthening effect and ductility compared to the fully bonded beam. Han et al. [10, 11] proposed a flexural analysis model of RC beam strengthened by the partially de-bonded FRP and demonstrated that the proposed model properly simulates the flexural behavior through an experiment. In particular, they confirmed that the ductility can be improved by the partial debonding. Their analysis method takes into account a change in the curvature distribution
ACCEPTED MANUSCRIPT according to the de-bonding and is based on the fact that FRP exerts sufficient anchorage performance at both ends of the beam. However, in this model, the anchorage behavior that occurs in the bonded part of the FRP strip was not adequately considered, but only the
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material properties of FRP were taken into account. Therefore, the bond characteristics of FRP strips by the de-bonding cannot be adequately taken into consideration.
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3. Flexural Behavior Model of RC beam strengthened by partially de-bonded NSM FRP
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3.1 Assumed condition
With regards to the NSMR using FRP in reinforced concrete flexural members, uniform deformation occurs as a whole in de-bonded FRP in the case of partial bonding at both ends and de-bonding in the central part. This leads to relief of a rapid increase of the deformation
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at a particular position in the case of full bonding, which is effective in relieving brittle failure phenomena. Fig. 1 is a picture showing the results of an experiment conducted by Seo et al. [24], and as shown in the figure, the failure is concentrated in the central part of the
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beam in the case of the full bonding, whereas cracks are distributed at appropriate intervals in the case of the de-bonding at the central part. It means that a slightly ductile behavior can be
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induced with an increase in the uniform deformation of FRP. However, in order to show such a behavior, FRP should extend a sufficient length to be anchored at ends. In order to represent the flexural behavior of the beam strengthened by partially debonded FRP, the following assumptions that are commonly defined in the reinforced concrete beams can be set forth as preconditions 1) Strain distribution of the section in the bonding area is linear. 2) There is no friction between concrete and FRP in the de-bonded region. 3) Moment is assumed to be linearly distributed with respect to the length.
ACCEPTED MANUSCRIPT 4) For concrete, a parabolic stress-strain relationship proposed by Collins and Mitchell [25] is used, and the concrete is assumed to be continuously deformed even after reaching strain at failure (Fig.2 (a)). 2
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ε ε fc f ' = 2 ⋅ c − c , ε c ' = 2 ⋅ c , Ec = 8500 ⋅ 3 f c ' fc ' εc' εc' Ec
(1)
5) Stress-strain relationship in the tension of concrete is linear and ignored after cracks
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occur. 6) Rebar is assumed to be a bi-linear model (Fig.2 (b)).
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7) Stress-strain relationship of the partially anchored FRP increases linearly up to the bond strength, and is assumed to be deformed until it reaches the maximum strain of FRP after reaching the bond strength (Fig. 2(c)).
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3.2 Bond strength of NSM FRP
As shown in Fig.3, if tensile force is applied to the FRP strip embedded in concrete, the failure modes according to the bond length can be divided into tensile failure strength of FRP
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( T ff ), epoxy bond failure strength ( T ft ) and bond failure strength ( T fb ). Since the tensile force of FRP in the flexural member is determined by the weakest among these three
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mechanisms, it can be summarized as follows.
Tf = min.{Tff , Tfc , Tfb } ·
(2)
T ff = φA f f fu
(3)
T fc = λτ ef (2b f Lb )
(4)
T fb = α p 0.85 n gηφ f
0.25
f c '0.33 ( L per γ )( E f A f )
(5)
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L γ = b L per
0.65
(6)
L γ + Be ≤ 1 .0 η = per nL γ per
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(7)
L per = 2b f + t f
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φ f = bf / t f
(8)
(9)
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Eq.(5) suggested by Seo [20] based on empirical formula of Sericiano et al. [19] considers the bond length effect and the group effect arising out of a compact space between FRP strips as shown in Fig.4. The resisting forces at FRP member anchorage according to bond length, based on Eqs.(2)-(9), are shown in Table 2. It was found that the bond failure strength ( T fb ) is
region.
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governing the failure for all case except CP1600-1 specimen fully bonded without de-bonded
3.3 Behavior analysis of RC beam strengthened in flexure by partially de-bonded NSM
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FRP strip
The flexural behavior of RC beam is normally explained by its curvature distribution at a level of load. Fig.5 represents the curvature distribution depending on de-bonding in the beam strengthened by the partial de-bonding. Until the central part reaches yield moment, the moment distribution shows a tendency to increase linearly from point ‘a’ to point ‘c’. In the case of fully bonded FRP, the curvature distribution shape is similar to that of moment. However, the curvature of RC beam with de-bonded region of FRP shows a drop at point ‘b’ where de-bonding begins since the strain of FRP is uniform in the de-bonded region. If the
ACCEPTED MANUSCRIPT moment in the central part is greater than the yield moment, the yield occurs within the debonded region. Therefore, the curvature distribution is divided into three sections such as elastic part in the bonded region, elastic part in the de-bonded region and inelastic part in the
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de-bonded region. Fig.6 represents the strain distribution of concrete on the compression side and FRP on the tension side after the central moment reaches the yield.
With an increase in the flexural moment, the curvature distribution for each step is
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calculated as in Fig.5, and the deflection at each position can be estimated. Fig.7 shows the strain distributions at the position ‘c’ in which the maximum stress occurs when the member
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reaches the yield moment and maximum moment, and ‘b’ section where de-bonding begins, respectively. Since point ‘b’ is the place where bonding ends and de-bonding begins, two kinds of deformation conditions are presented in the case of FRP. In other words, since the strain of FRP is the same in the de-bonded region, FRP strain of point ‘c’ becomes the same
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as that of point ‘b’, and therefore the strain of FRP cannot form a linear relationship with those of concrete and rebar at point ‘b’, but has a relatively higher value. In addition, an equilibrium relationship where the amount of deformation of de-bonded FRP is relatively less
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compared to that of tensile rebar even at point c where the maximum moment occurs should be considered to calculate the sectional forces.
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Fig.8 shows the procedures for calculating the flexural strength and deformation of the member corresponding to the increase of moment. The calculation procedures are as follows: i) on the base of the flexural theories, calculate the moment and curvature at point ‘b’ where de-bonding begins which corresponds to the moment of point ‘a’, the central part, and the moment of point ‘c’ if point ‘a’ exceeds the yield moment, ii) find the curvature at point ‘b’ by using the equilibrium relationship of the force since FRP strain within de-bonded region such as point ‘a’ and point ‘b’ is the same, iii) with the use of the above relationship, produce
ACCEPTED MANUSCRIPT the deflection by using the degrees of curves when the moment at point ‘a’ reaches the crack moment, yield moment and maximum moment, respectively.
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Step 1. Calculation of yield moment and curvature at perfect bond condition
The strengths of concrete, rebar and FRP at perfect bond condition can be calculated as
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follows:
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C c = 0 .5 f c ⋅ c y ⋅ b
C s = Asc ⋅ f sc Ts = As ⋅ f s
(11)
(12)
(13)
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Tf = Af ⋅ f f
(10)
In consideration of the equilibrium of forces in section, the neutral axis depth c y
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satisfying the following equation can be found.
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ΣPx = Cc + Cc − Ts − Tf = 0
(14)
With the use of c y , the yield moment and yield curvature are calculated as follows.
2 M y = Cc ⋅ (c y ) + Cs ⋅ (c y − d1 ) + Ts ⋅ (d − c y ) + T f ⋅ (d f − c y ) 3 when
ε s = ε sy ,
(15)
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(d − c y )
ε sy , ε sc =
(d − c y )
ε sy , ε f =
(d f − c y ) (d − c y )
ε sy
(16)
ε sy
(17)
M yb
M yb =
and
0.5L − Ldb ⋅My 0.5L
when the center reaches
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0.5L − Ldb ⋅ φy 0.5L
φyb
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(d − c y )
Step 2. Calculation of
φ y ,b =
(c y − d 1 )
My
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φy =
cy
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εc =
Step 3. Calculation of curvature φ y ,b − d at transit section when the center reaches
(18)
(19)
My
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FRP strain in de-bonded region is all the same, and it becomes the average value of the strains at section ‘b’ and ‘c’.
ε f ,b + ε f , c
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ε f ,b − d =
2
(20)
After calculating the stress and T f corresponding to the given FRP strain, the curvature at transit section is calculated by finding the compression fiber strain of concrete to match moment at the section and obtaining the neutral axis length c y which satisfies the equilibrium relationship of the section.
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φ y ,b − d =
ε cy ,b −d c y ,b − d
< φy
(21)
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Step 4. Calculation of the ultimate moment and curvature at perfect bond condition
The strength of concrete at perfect bond condition can be calculated with equivalent
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stress block as follows.
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C c = α 1 ⋅ β 1 ⋅ f c ⋅ cu ⋅ b when ε c = ε cu ,
ε cu cu
(cu − d 1 ) , ε s =
or when ε f =
ε fu ( d f − cu )
cu
cu , ε sc =
( d − cu ) , ε f =
ε fu
( d f − cu )
ε cu cu
( d f − cu )
( cu − d 1 ) , ε f =
ε fu ( d f − cu )
(23)
( d − cu )
(24)
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εc =
ε fu ,
ε cu
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ε sc =
(22)
The stress-strain relationship of FRP reflects the anchorage condition of FRP. The maximum moment and curvature are calculated by using cu that satisfies the equilibrium condition in section after finding it through iterative calculations.
M u = Cc ⋅
cu + C s ⋅ (cu − d1 ) + Ts ⋅ (d − cu ) + T f ⋅ (d f − cu ) 2
(25)
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φu =
εc
(26)
cu
In relation to the maximum moment estimation, although the strength of member was
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calculated at perfect bond condition, the anchorage condition of the end was taken into account, and therefore the strength of the case where the anchorage is not sufficient in partial de-bonding can be properly calculated.
0.5L − Ldb ⋅ φu 0.5L
(27)
(28)
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φu ,b =
0.5L − Ldb ⋅ Mu 0.5L
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M u ,b =
φu ,b when the center reaches M u
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Step 5. Calculation of M u ,b and
Step 6. Finding the location of yielding section when the center reaches M u
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M Luy = 0.5 L1 − y Mu
Step 7. Calculation of curvature at transit section,
(29)
φu ,b−d when the center reaches M u
FRP strain in de-bonded region is all the same, and it becomes the average value of the strains at section ‘b’ and ‘c’. The value is to be its ultimate strain ε fu when the FRP tension failure governs.
ACCEPTED MANUSCRIPT After calculating the stress and T f corresponding to the given FRP strain and obtaining the neutral axis depth cu which satisfies the equilibrium relationship of the section by using
ε cu ,b − d c u ,b − d
< φu
(30)
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φ u ,b − d =
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ε c ,b = ε c ,b −d , the curvature at transit section is calculated.
the relationship of
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3.4 Analysis result and discussion
According to the aforementioned analysis procedures, calculations were performed with specimens of Seo et al. [24], and the adequacy of the analysis method was compared. The details of the specimens are shown in Fig.9 and Table 1. Table 2 shows the resisting force of FRP anchorage part depending on the bond lengths calculated using Eq.(2) ~ (10). Fig.10
experimental results.
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represents a comparison of load-displacement curve between analysis results and
The analysis results of the case where the entire lower part is strengthened by NSM FRP
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strip without de-bonded region showed that concrete on the compression side reached the
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maximum strength before FRP reached the maximum strain. In the experiment, the strength increased almost linearly until the concrete on the compression side reached the maximum strength, forming a high yield point. Moreover even after concrete on the compression side reached the maximum strength, it exerted a predetermined capacity with no rapid decrease in the internal force until FRP on the tension side suffered a fracture. This is because as a plate for loading with width of 50mm confined concrete on the compression side to some extent in the place where the compressive stress is concentrated, concrete was sufficiently resistant to compression without reaching brittle failure rapidly (refer to Fig. 11 failure mode).
ACCEPTED MANUSCRIPT Accordingly, the analysis assumed that concrete on the compression side maintains the maximum strength until FRP reaches its ultimate strain even after reaching the maximum strength, and therefore the maximum moment and curvature at fracture of FRP on the tension
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side were represented together. In the case of specimens with de-bonded region of FRP in central part, the maximum internal forces were found to be determined by the fracture of FRP anchorage part on the
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tension side. Although the stiffness until the yield was slightly high, and the amount of deflection until the strengthening effect is lost was rather low in the analysis results, the
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analysis results are in good correspondence with experimental results as a whole. Fig.12 shows the curvature distribution when the central part of member reaches the maximum strength and yield, and Fig.13 shows the strain of the compressive edge, compressive rebar, tensile rebar, and FRP at each position when the central section reaches the maximum
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moment. Since the strain distribution is linear in the bonding area, but FRP strain is uniform in the de-bonded region, the strain of FRP is relatively larger than that of tensile rebar at point ‘b’ but smaller at point ‘c’.
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With respect to the maximum internal force and yield of the member, a comparison between experimental results and analysis results is shown in Table 3. The stiffness until the
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yield was slightly high, and the amount of deflection until the strengthening effect is lost was rather low in the analysis results. Fig.14 represents the changes in load-displacement relationship due to an increase in de-bonded length, and as the de-bonded length increases, the deformation capacity is improved after yielding.
4. Conclusion
ACCEPTED MANUSCRIPT In this paper, a simple analytical method describing the flexural behavior of reinforced concrete beams strengthened with partially de-bonded NSM FRP strips has been studied. Especially, the characteristics of anchorage part and de-bonded region were reflected in the
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analysis. In this regard, in this study, it has proposed a method for flexural analysis of the partially de-bonded FRP strip by reflecting the bond behavior of NSM FRP strip in the flexural model based on the stress transfer mechanism between NSM FRP strip and concrete.
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Some concluding remarks are listed as follows.
The strain of FRP at the de-bonded region was assumed to be uniform, and the
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equilibrium of the section was satisfied with respect to the applied load. Therefore, it was found that the flexural behavior of the reinforced concrete beam strengthened by partially debonded NSM FRP strip can be described.
According to the length of the de-bonded region, failure mechanisms can be divided into
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tensile failure of FRP, tensile failure of concrete, and bond failure of epoxy, of which the dominant failure mechanism were taken into consideration to conduct flexural analysis of the case being dominated by the behavior of FRP.
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The analysis results of the case where the entire lower part is strengthened by NSM FRP strip without de-bonded region revealed that concrete on the compression side showed brittle
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failure mode after reaching the maximum strength. However, in the actual experiment, concrete on the compression side exerted a predetermined capacity with no rapid degradation of the internal force even after reaching the maximum strength, and thus there is a need to reflect this characteristic in the actual analysis.
Acknowledgement This research was supported by a grant from the Academic Research Program of Korea
ACCEPTED MANUSCRIPT National University of Transportation in 2013.
Notation
bf ⋅ t f
(mm2)
= cross-sectional area of FRP reinforcement =
As , Asc
= cross-sectional area of tension bar and compression bar, respectively
Be
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(mm2)
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Af
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= distance between centroids of FRP strips located at extreme sides in considering group effect
= width of beam and NSM FRP strip, respectively (mm)
Cc , C s
= resultant compression force of concrete and steel reinforcement,
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b , bf
respectively (N)
= neutral axis depth at yield and ultimate section, respectively (mm)
c y ,b , cu ,b
= neutral axis depth of bonded part at transit section when the beam
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c y , cu
researches yield and ultimate state, respectively (mm)
c y ,b−d , cu ,b −d
= neutral axis depth of de-bonded part at transit section when the beam researches yield and ultimate state, respectively (mm)
d , d1 , d f
= depth of tension steel reinforcement, compression steel reinforcement and FRP reinforcement, respectively (mm)
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= modulus of elasticity of concrete and FRP reinforcement, respectively (Mpa)
f c , fc '
= compressive stress and ultimate compressive strength of concrete,
f f , f fu
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respectively (Mpa)
= tensile stress and ultimate tensile strength of FRP reinforcement,
= stress of steel reinforcements at compression and tension, respectively (Mpa)
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f sc , f s
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respectively (Mpa)
= beam span (mm)
Lb , L db
= bonded length and de-bonded length of FRP reinforcement, respectively (mm)
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L
= circumferential length of FRP (mm)
Luy
= length from loading point to yielding section (mm)
M y ,M u
= yield and ultimate moment of beam, respectively (N.mm)
Mb
= moment at transit section (section ‘b’) (N.mm)
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L per
M y ,b , M u , b
= moment at transit section when the center reaches yield and ultimate moment, respectively (N.mm)
ng
= number of FRP strip in considering group effect
ACCEPTED MANUSCRIPT tf
= thickness of FRP strip (mm)
Ts , Tf
= resultant tension force of steel reinforcement and FRP reinforcement,
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respectively (N) = ratio of average stress in rectangular compression block
αp
= test coefficient (0.7 is used for the calculation of flexural strength of beam
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α1
in this study).
= ratio of depth of rectangular compression block to the neutral axis
εc ,εc '
= concrete strain at certain state and ultimate strength, respectively
ε c ,b , ε c ,b − d
= concrete strain of bonded part and de-bonded part at transit section,
ε cy , ε cu
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respectively
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β1
= strain at yield and ultimate state of beam, respectively
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ε cy ,b−d , ε cu ,b−d = concrete strain at transit section when the center reaches yield and ultimate
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state, respectively
ε f , ε fu
= strain of FRP reinforcement at certain level and ultimate strength, respectively
ε f , b , ε f ,b − d
= strain of FRP reinforcement of bonded part and de-bonded part at transit section, respectively
ε fy,b , ε fu ,b
= strain of FRP reinforcement of bonded part at transit section when the
ACCEPTED MANUSCRIPT center reaches yield and ultimate state, respectively
ε fy ,b−d , ε fu ,b−d
= strain of FRP reinforcement of de-bonded part at transit section when the center reaches yield and ultimate state, respectively = strain of FRP reinforcement at section ‘c’
ε fy
= strain of FRP reinforcement at yield state
ε s , ε sc
= strain of steel reinforcement in tension part and in compression part,
ε sy,b−d , ε su ,b−d
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respectively
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ε f ,c
= strain of steel reinforcement in tension part when the center reaches yield and ultimate state, respectively
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ε scy ,b−d , ε scu ,b−d = strain of steel reinforcement in compression part when the center reaches yield and ultimate state, respectively = yield strain of steel reinforcement
φ
= strength reduction factor (1.0 is used in this study) (rad/mm)
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φ y , φu
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ε sy
φu ,b , φu ,b−d
= curvature of beam at yield and ultimate moment, respectively (rad/mm)
= curvature of beam at the bonded and de-bonded part of transit section when the center reaches ultimate state, respectively (rad/mm)
φ y , b φ y ,b − d
= curvature of beam at the bonded and de-bonded part of transit section when the center reaches yield state, respectively (rad/mm)
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λ
= effective bond loss factor (1.0 is used in this study)
τ ef ,τ f
= shear strength of epoxy and bond strength of NSM FRP strip, respectively
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(Mpa)
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References [1] D’Ambrisi A., Feo L., Focacci F. Experimental analysis on bond between PBO-FRCM strengthening materials and concrete. Compos Part B Eng 2013; 44:524- 532.
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[2] Ascione L., Feo L. Modeling of composite/concrete interface of R/C beams strengthened with composite laminates. Compos Part B Eng 2000;31: 535-540.
[3] Sharma SK, Mohamed Ali MS, Goldar D, Sikdar PK. Plate-concrete interfacial bond
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strength of FRP and metallic plated concrete specimens. Compos Part B Eng 2006; 37: 54-63.
Part B Eng 2005; 36(2): 99-113.
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[4] J Yao, JG Tang, JF Chen. Experimental study on FRP-to-concrete bonded joints. Compos
[5] D’Ambrisi A., Feo L., Focacci F. Bond-Slip Relations for PBO-FRCM Materials Externally Bonded to Concrete. Compos Part B Eng 2012; 43: 2938- 2949
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[6] Ascione L., Berardi V., FEO L., Mancusi G. A numerical evaluation of the inter laminar stress state in externally FRP plated RC beams. Compos Part B Eng 2005;36: 83-90. [7] D'Ambrisi A., Feo L., Focacci F.
Experimental and analytical investigation on bond
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between Carbon-FRCM materials and masonry. Compos Part B Eng 2013:46:15- 20. [8] Yuan H., Lu X., Hui D., Feo L. Studies on FRP-concrete interface with hardening and
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softening bond-slip law. Compos Struct 2012;94:3781- 3792. [9] Aprile A., Fe L., Concrete cover rip-off of R/C beams strengthened with FRP composites. Compos Part B Eng 2007; 38:759-771. [10] Han TC, Jeffrey SW, Khaled AS. Partially bonded near-surface-mounted CFRP bars for strengthened concrete T-beams. Construction and Building Materials 2011;25:2441-2449. [11] Han TC, Jeffrey SW, Khaled AS. Analysis of the Flexural Behavior of Partially Bonded FRP Strengthened Concrete Beams, Journal of Composites for Construction 2008;12(4):375-386.
ACCEPTED MANUSCRIPT [12] Seo S, Yoon S, Kwon Y, Choi K. Bond behavior between near surface-mounted fiber reinforced polymer plates and concrete in structural strengthening. Journal of Korea Concrete Institute 2011;23(5):675-82 (written in Korean).
concrete structures, Composite Structures 2013;95:719–727.
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[13] Seo S, Feo L, Hui D. Bond strength of near surface-mounted FRP plate for retrofit of
[14] Barretta R., Luciano R., Willis J.R. On torsion of random composite beams. Composite
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Structures 2015;132: 915-922.
[15] Barretta R., Feo L., Luciano R. Some closed-form solutions of functionally graded beams
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undergoing nonuniform torsion. Composite Structures 2015;123:132-136. [16] Barretta R., Luciano R. Analogies between Kirchhoff plates and functionally graded Saint-Venant beams under torsion. Continuum Mechanics and Thermodynamics 2015;27:499-505.
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[17] Barretta R., Feo L., Luciano R. Torsion of functionally graded nonlocal viscoelastic circular nanobeams. Composites: Part B 2015;72:217-222. [18] Apuzzo A., Barretta R., Luciano R., Some analytical solutions of functionally graded
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Kirchhoff plates. Composites: Part B 2015;68:266-269. [19] Seracino R, Jones N, Ali M, Page M,. Oehlers D. Bond strength of near-surface mounted
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FRP strip-to-concrete joints. Journal of Composite Construction.2007;11(4):401–409. [20] Seo S. Bond Strength of Near Surface-Mounted FRP Strip in RC Member. Journal of the Korea Concrete Institute 2012; 24(4):415-422 (written in Korean). [21] Lees JM, Burgoyne CJ. Experimental study of influence of bond on flexural behavior of concrete beams pretensioned with aramid fiber reinforced plastics. ACI Struct J 1999;96(3):377–85. [22] Lees JM, Burgoyne CJ. Analysis of concrete beams with partially bonded composite
ACCEPTED MANUSCRIPT reinforcement. ACI Struct J 2000;97(2):252–60. [23] Ali C, Khaled S. Flexural Response of Reinforced Concrete Beams Strengthened with End-Anchored Partially Bonded Carbon Fiber-Reinforced Polymer Strips, Journal of
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Composites for Construction 2005: 9(2):170-177. [24] Seo S, Choi K, Kwon Y, Lee K. Flexural Strength of RC Beam Strengthened by Partially
and Materials 2016; Online published.
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De-bonded Near Surface-Mounted FRP Strip, International Journal of Concrete Structures
[25] Collins MP, Mitchell D. Prestressed concrete basics. Canadian Prestressed Concrete
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Institute (CPCI), Ottawa, Ontario; 1987.
Table 1 Specimen list
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Specimen name
Retrofit method
Bonded length (mm)
De-bonded length (mm)
BC2000
None
-
-
1,600
-
500 at each ends
600
300 at each ends
1000
CP1600-1
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3.6×16
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CP300-1
NSMR
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CP500-1
Strip dimension (width×height) (mm×mm)
ACCEPTED Table 2 Internal tensile forces of NSM FRP strip MANUSCRIPT Forces corresponding to the bonded length (kN)
Forces
Lb=400mm
Lb=500mm
Lb=800mm
T ff
161.28
161.28
161.28
161.28
T ft
172.80
230.40
288.00
460.8
T fb
85.44
93.81
100.87
235.02
Tf
78.07
90.12
103.62
161.28
T f / T ff
0.53
0.58
0.63
1.00
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Lb=300mm
Table 3 Comparison with test and analysis results
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Specimen
BC2000
Yield
Ultimate
Test or Analysis
Py (kN)
δ y (mm)
Ke
Pu (kN)
δ u (mm)
Test
143.18
6.03
23.74
190.24
79.61
Analysis
142.5
4.38
32.53
159.03
19.27
Test
200.55
6.72
29.84
225.55
15.98
Analysis
150.00
4.11
36.50
215.96
10.78
Test
161.32
6.35
25.40
208.39
19.06
Analysis
150.00
3.94
38.07
205.94
15.73
Test
154.95
6.63
23.37
194.17
19.16
Analysis
150.00
3.93
38.17
198.04
17.08
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CP300-1
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CP500-1
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CP1600-1
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Fracture is concentrated on the beam center
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(a) Fully bonded NSM FRP along the beam length
(b) Partially de-bonded NSM FRP around center of beam
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Fig. 1 Spreading out of cracks by fully bonded and partially de-bonded NSM FRP
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(a) Concrete
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(b) Reinforcement
(c) Bond stress-strain relationship of NSM FRP embedded in concrete Fig. 2 Stress-strain curves of materials
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Fig. 3 Internal force of partially de-bonded NSM FRP
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Fig. 4 Effective concrete width of NSM FRP strips
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Fig. 5 Moment and curvature distribution of RC beam with partially de-bonded NSM FRP
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Fig. 6 Strain distribution of partially de-bonded NSM FRP in strengthened beam
(a) Yield condition
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(b) Ultimate condition
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Fig. 7 Strain distributions of a section at yield and ultimate condition
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Fig. 8 Calculation process of the deflection of RC beam with partially de-bonded NSM FRP
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Fig. 9 Detail of the specimen done by Seo et al.
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(b) Fully bonded
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(a) RC beam w/o NSM FRP
(c) Lb = 500mm
(d) Lb =300mm
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Fig. 10 Comparison of calculation and test results in load-displacement
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Fig. 11 Failure shape of CP1600-1 specimen
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(a) Fully bonded
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(b) Lb=500mm
(c) Lb = 300mm
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Fig. 12 Curvature distribution at yield and ultimate state
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(b) at transit section (point b), de-bonded region
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(a) at transit section (point b), bonded region
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(c) at yield section
(d) at center section (point c)
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Fig. 13 Distributions of FRP strains at ultimate condition (Lb=300mm)
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Fig. 14 Load-displacement variation corresponding to the de-bonded length