Effective Strain of RC Beams Strengthened in Shear with FRP

Transcription

1 Eective Strain o RC Beams Strengthened in Shear with FRP Author Lee, Jung-Yoon, Hwang, Hyun-Bok, Doh, Jeung-Hwan Published 212 Journal Title Composites Part B: Engineering DOI Copyright Statement 211 Elsevier. This is the author-manuscript version o this paper. Reproduced in accordance with the copyright policy o the publisher. Please reer to the journal's website or access to the deinitive, published version. Downloaded rom Griith Research Online

2 Accepted Manuscript Eective Strain o RC Beams Strengthened in Shear with FRP Jung-Yoon Lee, Hyun-Bok Hwang, Jeung-Hwan Doh PII: S (11)519-1 DOI: 1.116/j.compositesb Reerence: JCOMB 1598 To appear in: Composites: Part B Received Date: 29 October 21 Revised Date: 19 July 211 Accepted Date: 3 November 211 Please cite this article as: Lee, J-Y., Hwang, H-B., Doh, J-H., Eective Strain o RC Beams Strengthened in Shear with FRP, Composites: Part B (211), doi: 1.116/j.compositesb This is a PDF ile o an unedited manuscript that has been accepted or publication. As a service to our customers we are providing this early version o the manuscript. The manuscript will undergo copyediting, typesetting, and review o the resulting proo beore it is published in its inal orm. Please note that during the production process errors may be discovered which could aect the content, and all legal disclaimers that apply to the journal pertain.

3 Eective Strain o RC Beams Strengthened in Shear with FRP Jung-Yoon Lee a, Hyun-Bok Hwang a and Jeung-Hwan Doh b* a Department o Civil, Architectural and Environmental System Engineering, Sungkyunkwan University, Suwon, Republic o Korea b Griith School o Engineering, Griith University Gold Coast Campus, Queensland 4222, Australia (corresponding author, j.doh@griith.edu.au) ABSTRACT The ailure modes o reinorced concrete (RC) beams strengthened in shear with iber reinorced polymer (FRP) sheets or strips are not well understood as much as those o RC beams reinorced with steel stirrups. When the beams are strengthened in shear with FRP composites, beams may ail due to crushing o the concrete beore the FRP reaches its rupture strain. Thereore, the eective strain o the FRP plays an important role in predicting the shear strength o such beams. This paper presents the results o an analytical and experimental study on the perormance o reinorced concrete beams strengthened in shear with FRP composites and internally reinorced with conventional steel stirrups. Ten RC beams strengthened with varying FRP reinorcement ratio, the type o iber material (carbon or glass) and coniguration (continuous sheets or strips) were tested. Comparisons between the observed and calculated eective strains o the FRP in the tested beams ailing in shear showed reasonable agreement. Keywords: A. Polymer Fibre; B. Strength; B. Mechanical Properties; D. Mechanical Testing. * Corresponding author. Tel.: ; Fax: address: j.doh@griith.edu.au (J.-H. Doh).

4 1. Introduction Fiber Reinorced Polymer (FRP) is a relatively new class o composite material manuactured rom ibers and resins and has proven eicient and economical or the development and repair o deteriorating structures in civil engineering. The strength o FRP-strengthened RC members is inluenced by the type o ibers, iber directions, iber distributions, and bond schemes. The eectiveness o FRP is maximized by bonding the external FRP reinorcement parallel to the direction o principal tensile stress. FRP bond schemes (e.g. side bonding, U jacketing, wrapping) can also result in dierent shear capacity o FRP-strengthened RC members. Detailed investigations on the shear strengthening o RC members have been relatively limited, and many questions regarding the strengthening mechanism are yet to be resolved. With a ew exceptions, most researchers have idealized the FRP materials as being analogous to internal steel stirrups, assuming that the contribution o FRP to shear capacity emanates rom the capacity o the ibers to carry tensile stresses at a more or less constant strain, which is equal to or lower than the FRP ultimate tensile strain. Current national code provisions are available or designing elements externally bonded with FRP, and many experimental tests have been carried out in this ield o research. Both experimental evidences and code suggestions indicate that a perormance improvement can be realized, but even i a lot o solutions are possible, their perormances and beneicial eects are not yet quantiied [1].

5 Table 1 shows various equations (see Eq. 1 to 5) developed by others to estimate the e eective strain o FRP ( ε ) at shear ailure. Recently, Teng et al. [2] tested nine FRPstrengthened RC beams: three as control specimens, three with bonded FRP ull wraps, and three with FRP ull wraps let unbonded to the beam sides. The test results showed that the unbonded FRP wraps had a slightly higher shear strength capacity than the bonded FRP wraps. In addition, Deialla and Ghobarah [3] tested six hal-scale beams externally bonded with FRP composites constructed using a specially designed test setup in order to investigate the strengthening techniques or beams subjected to combined shear and torsion. Test results indicated that externally bonded CFRP strengthening schemes enhanced the shear and torsion carrying capacities o RC beams. Grande et al. [4] tested iteen RC beams strengthened in shear by externally bonded iber-reinorced plastics (FRP) sheets to study the inluence that the geometrical percentage o transverse steel reinorcement could had on the FRP resisting action. The experimental investigation indicated the variability o the FRP shear resisting action over the amount o the transverse steel reinorcement. In particular, the FRP shear resisting action was generally smaller in beams with closer stirrups. Fico et al. [5] studied the assessment o Eurocode design equations or the evaluation o the shear strength o FRP RC members, as proposed by the guidelines o the Italian Research Council. Li et al. [6] tested sixteen RC beams with or without FRP composites to study the shear strengthening eect. The experimental results indicated that the shear comtribution o FRP composites was

6 inluenced by the applied FRP composite area, the spacing between the steel stirrups, and the longitudinal steel bar diameter o the RC beams. Although there are many proposed equations to determine ε e, it is generally agreed that the eective strain o FRP is diicult to predict solely by rigorous analysis because ε e is inluenced not only by matierial properties but also by the shear ailure modes o strengthened RC members. This research aims to investigate the eective shear strain o the FRP used as shear strengthening material in beams that show two dierent shear ailure modes. Included in the study are the stresses and strains o concrete, steel stirrup, and FRP at shear ailure. 2. Research signiicance The shear ailure modes o FRP strengthened concrete beams are quite dierent rom those o the beams strengthened with steel stirrups. This paper presents the results o an experimental and analytical study on the perormance o reinorced concrete beams externally wrapped with FRP composites and internally reinorced with steel stirrups. In this investigation, emphasis was placed on the eective strain o the materials (concrete, FRP, and steel bars) o the FRP shear strengthened RC beams at peak load (ultimate stage). 3. Eective strain o the FRP 3.1 Failure modes o RC beams shear-strengthened with FRP composites

7 The ailure modes o RC beams shear-strengthened with FRP composites are more complex than those o RC beams reinorced with conventional steel stirrups. Debonding o FRP composites or concrete crushing, beore yielding o the steel stirrups, cause brittle ailure. On the other hand, concrete crushing or FRP rupture, ater yielding o the steel stirrups, cause ductile ailure modes. In this paper, we deine the two latter ailure modes as: Failure mode 1: concrete crushing ater yielding o the steel stirrups. Failure mode 2: rupture o FRP composites ater yielding o the steel stirrups without concrete crushing. For the beams showing ailure mode 1, the shear resistance contribution o FRP ( V ) should be calculated by using the eective strain o FRP ( ε ), because the stress o the FRP at shear ailure does not reach its rupture strength ( e r ). However, or the beams exhibiting ailure mode 2, the stress o the FRP at shear ailure can be replaced by its rupture strength. FRP debonding ailure is one o the main ailure modes or RC beams strengthened in shear with externally bonded FRP. The separate treatment o FRP debonding and FRP rupture is essential to develop accurate shear strength models because the ailure mechanism o FRP debonding is quite dierent rom that o FRP rupture [1]. This paper placed emphasis on presenting a model to predict the eective strain o RC beams shear strengthened with FRP wraps ailing in Failure mode 1 or 2 without debonding.

8 3.2 FRP ratio at the balanced shear ailure The ollowing is the most widely used expression to calculate the shear capacity o FRPstrengthened RC beams, V n : Vn = Vc + Vs + V (1) where V c, V s, and V are the shear resistance contributions o concrete, steel stirrups, and FRP, respectively. Based on the truss model, the shear resistance contribution o FRP can be derived as ollow, V = ε E ρ b d(cotθ + cot α)sinα (2) e w where ε e is the eective strain o FRP when a RC member reaches its shear capacity; E is the elastic modulus o FRP; ρ is the FRP reinorcement ratio; bw is the minimum width o cross section; d is the eective depth o cross section; θ is the diagonal crack angle; and α is the angle between principal iber orientation and longitudinal axis o the member. As indicated by the test results in the literatures, the strain distributions o the FRP composites were not uniorm and the strain concentrations occurred near the locations o diagonal cracks. In this study, the assumption o continuous material was applied to the compatibility aided truss model to get the eective strain o FRP because the stress (or strain) distributions o the FRP composites are not uniorm. Thus, all the stresses and strains involved in this model are the average (or smeared) stresses and average (or smeared) strains. The eective strain o the FRP at shear ailure was related to the amount o shear

9 reinorcement and the ailure mode. In this paper, the eective strain o the FRP at shear ailure was ormulated starting with the FRP reinorcement ratio corresponding to a balanced shear ailure. Fig. 1 shows the stress conditions o the FRP strengthened RC beams ater the ormation o diagonal cracks. Based on the well-known variable-angle truss models in which the angle θ o the cracks in concrete is assumed to be equal to the angle o the principal compressive stress o concrete, the tensile orce ρ tv tv in the stirrup must counteract the vertical component o the diagonal compressive strut under the equilibrium conditions, and the required FRP reinorcement ratio can be estimated as: 2 2 ( 2sin 1cos ) 1 ρ = θ θ ρtv tv (3) sinα where 1 and 2 are the principal tensile and compressive stresses o concrete, respectively; ρ tv and ρ are the steel ratio in the transverse direction and the FRP ratio, respectively; tv and are the steel stress in the transverse direction and the stress o the FRP, respectively. Fig. 2 shows two typical examples o beam responses up to ailure. The horizontal axis indicates the shear strain. The vertical axis in the upper direction indicates stress o the FRP composites and stress o the steel bars, while the vertical axis in the lower direction indicates principal compressive stress o concrete ( 2 ) and the eective compressive strength o concrete ( v ' ). For the beams with the majority o FRP composites showing ailure mode 1, c the concrete crushes beore the stress o the FRP reaches its rupture strength ( < r and

10 2 = v c '), while or the beams with smaller amount o FRP composites showing ailure mode 2, the FRP ruptures beore concrete crushing ( = r and 2 < v c ' ). The boundary between Failure mode 1 and 2 represents the balanced shear ailure in which the concrete crushes at the same time as the FRP reaches its rupture strength ater the steel stirrups yielded. In the balanced shear ailure mode, the principal compressive stress o concrete 2 in Eq. (3) must equal the eective compressive strength o concrete v c ', and the tensile stresses tv and must be equal to the yield strength tvy and the rupture strength r, respectively. Fig. 3 shows the Mohr's stress and strain circles at the boundary between ailure mode 1 and ailure mode 2. The principal tensile stress o concrete 1 is ignored because it is much smaller than the other strengths ' v c, tvy, and r. Thereore, the FRP ratio ( ρ b ) at the balanced shear ailure can be ormulated as: 2 ( v 'sin ) 1 ρb = c θ ρtv tvy (4) sinα r where ρ b is the FRP reinorcement ratio required or a balanced shear ailure. The angle θ o the diagonal cracks in Eq. (4) can be determined using Mohr s strain circle as shown in Fig. 3(c). ε l and ε t are the normal strains in the longitudinal and transverse directions; ε 1 and ε 2 are the strains o concrete in the principal tension and compression directions, and γ is the shear strain. Thus, the angle θ can be expressed in terms o the strains as: ( ) ( ) cos θ = ε / ε + ε ε (5) t l t 2

11 At the balanced shear ailure, the strains o the steel stirrups and the FRP are equal to the rupture strain ε r o the FRP. In addition, because the principal compressive strain o concrete ε 2 is much smaller than ε r and ε l, the principal compressive strain o concrete can be ignored in Eq. (5) resulting in the ollowing equation: ( r) ( l r) cos θ = ε / ε + ε (6) Several equations [11-14] gave the constitutive relationships o cracked concrete in the principal compressive and tensile directions o concrete. In this study, the eective compressive strength o concrete proposed by Collins et al. 7) will be adopted as below: 1 v ' = c c ' ( ε + ε ) (7) The eective compressive strength coeicient o concrete v o Eq. (7) was originally l r ormulated to predict the shear strengths as well as the deormations o RC membrane elements, such as panels, subjected to shear and normal loads. The equation was also utilized to predict the shear behavior o RC beams [16, 17]. The equilibrium conditions or the longitudinal stresses must satisy the ollowing Eq. (8): = cos θ + (1 cos θ) + ρ E ε + ρ cosα (8) l s l r By substituting Eq. (6) into Eq. (8) and assuming thatε 2 = 1 = or the reasons mentioned earlier, the strain o the tensile longitudinal steel bars ε l can be obtained using Eq. (9): l ε r ε + ε r ( ) v ' + ρ E ε + ρ cosα = c l s l r (9) The FRP balanced shear reinorcement ρ b r was calculated by substituting v c ' and

12 θ in Eq. (4). The FRP balanced shear ratio vs. the compressive strength o concrete curves or a RC beam calculated using Eq.(4). It is noted that the balanced shear ratio o the FRP ρ b increases as the compressive strength o concrete ' c increases. This is because as the rupture strain o FRP increases, the strain o concrete in the principal tension direction ε 1 increases. Accordingly, the ρ b is correspondingly reduced due to the decrease in the eective compressive strength o concrete v c ' that decreases with the increase o ε Calculation o the eective strain o the FRP As mentioned in the early part o this paper, or the beams showing ailure mode 1, the concrete crushes beore the stress o the FRP reaches its rupture strength. Thereore, the eective strain o FRP at shear ailure can now be determined by replacing ρ b r with ρ (= E ρ ε ) in Eq. (4). For the ailure mode 1 beams, For the ailure mode 2 beams, 2 ( 'sin ) ε eρe sin α = vc θ ρtv tvy ( ρ ρ b ) (1a) ε e = ε r ( ρ ρ b < ) (1b) In Eq.(1a), v c ' and θ can be calculated by replacing ε r with ε in Eqs. (7) and (6), respectively. The amount o the FRP vs. the strain o the FRP curves or a RC beam are shown in Fig. 4 in which the solid line represents the FRP reinorcement ρ, calculated

13 using the right term o Eq.(1a), while the dotted line calculated using the let term o Eq. (1a) that linearly increases as the strain o the FRP ( ε ) increases. The intersecting point where two lines cross is the eective strain o FRP at shear ailure calculated by Eq.(1a). In case o a beam having balanced shear FRP ratio ρ and ' = 4MPa (Beam 1, b c ρ = ρ =.244, = 3,825MPa, ε =.153, E = 25GPa, β = 9 ), two lines b r r cross at point A where the calculated strain is.153 that is the rupture strain o the FRP. I the beam is strengthened with a great amount o the FRP than required or the balanced shear ailure, say ρ =.8, the slope o the dotted line increases and the eective strain o the FRP ε e, becomes.1 o the value at point B (Beam 2). Thereore, this beam ails due to crushing o the concrete beore the FRP reaches its rupture strain because the eective strain o.1 at point B is smaller than the rupture strainε r =.153. Similarly, i the FRP reinorcement ratio ρ becomes.16, the calculated eective strain o the FRP ε e, decreases to ε =.77 o point C as shown in Fig. 4 (Beam 3). It is noted that the eective strain o the FRP ε e decreases as the FRP reinorcement ratio ρ increases or the equilibrium o compressive orce o concrete and tension orce o FRP composites. The material properties and analytical results o these three example beams are listed in Table 2. To be able to deine the shear strength o RC beams, the eective compressive strength coeicient o concrete v, the normal strain in longitudinal direction ε l, and the angle θ o the diagonal cracks should be calculated rom Eqs. (6), (7), and (9), respectively. This will

14 then allow the eective strain o the FRP, ε e, to be determined using Eq. (1). Eq. (2) is then used to calculate the shear resistance contributions o FRP, V. 4. Test program and measurements 4.1. Test specimens A total o 1 RC beams with the same cross-sectional dimensions were strengthened by FRP-wrap and tested to investigate the eectiveness o FRP at shear ailure in detail. The proposed equation (1) is applicable to predict the eective strain o the FRP in RC beams that ail due to crushing o the concrete or FRP rupture without debonding (or premature ailure/ peeling o). Thereore the test beams were strengthened by FRP wrapping and designed to prevent debonding ailure. The overall dimensions, arrangement o reinorcement, and loading system are shown in Fig. 5. Three parameters were considered in this investigation: FRP reinorcement ratio, the type o iber materials (carbon C or glass G ) and coniguration (continuous sheets C or strips S ). All beams were reinorced with ten bottom 25.4mm steel bars (D25 steel bar) and ive top 25.4mm steel bars. The longitudinal tensile reinorcement ratio o all beams was.367. The shear reinorcement consisted o 9.53mm steel stirrups (D1 steel bar) spaced each 2mm. The ratio o steel stirrups was.24. The yield strengths o D25 and D1 steel bars were MPa and 3. MPa, respectively. All beams were designed to share the same shear span-to-depth ratio ( a/ d =3).

15 The specimens were cast in ormworks o which our corners were rounded at radius o 1mm. These round corners were necessary to alleviate stress concentraion which causes premature ailure o the external FRP. The beams were removed rom the orms ater 72 hours. The beams were cured at room temperature or a suicient period. Ater curing the beams, the suraces o the specimens were cleaned using an electric grinder, and primer was applied to the suraces o the beams. The FRP sheets were then placed using the epoxy (Epondex by Hankuk Carbon Co. Ltd.) ater the primer had completely dried. One concrete batch was used to cast all the beams at the same time. Concrete cylinders were tested on the irst and the last day o beam testing. Each test consisted o three cylinders. The average concrete strength o the two cylinder tests was 35.MPa. Speciications o specimens are given in Table 3. Mechanical properties o the FRP composites and the longitudinal steel rebars used in this investigation are shown in Table Loading system and measurements The specimens were simply supported and were subjected to a concentrated load at midspan. Electrical strain gages were installed on the suraces o the steel bars and FRP sheets to record the strains during the test, as shown in Fig. 5. Because the strain gages can measure only one point strain o stirrups or FRP sheets, ive linear displacement transducers (LVDTs) were attached to each ace o the beam near the shear critical region to measure the

16 longitudinal, transverse, and shear average strains o each region. It is noted that the eective strain in Eq.(1) is not the maximum strain but the average strain derived rom the compatibility aided truss model. An additional LVDT was placed under the beam at mid-span to measure the delection o the beam. 5. Test Results All beams ailed in shear beore lexural yielding, except beam C2 which ailed in shear ater lexural yielding. Premature ailure due to local racture or debonding o the FRP did not occur in any o the specimens. In the beams strengthened with FRP sheets (beams in C- or G- series), obvious diagonal cracks were not observed at a lower load level because o the presence o the FRP sheets. As the applied load increased, part o the FRP close to the diagonal cracks was torn apart, producing noise. In some specimens, partial debonding o the FRP occurred near the critical shear crack or on the edge o the beam though, eventual ailure occurred due to crushing o the concrete. At onset o the concrete ailure, strain measurements by strain gages and LVDTs indicated that the FRP did not reach its ull tensile capacity Load-delection curves The beams strengthened with glass-frp strips showed lexural cracks near the bottom surace o the beam close to mid-span. The ormation o these cracks was also relected in the

17 load-delection curve as a sudden reduction in the stiness o beams. The test results including the shear strength o the beam and the ailure modes are shown in Table 3. The load-delection curves o the 1 RC beams are presented in Figs. 6 (a) through (c). The strengths o the beams increase with an increasing amount o FRP sheets and strips. Enhanced behaviors o FRP strengthened beams are not only observed in the strength but also in the delection corresponding to the peak load, leading to higher energy dissipation capacity. As an example, the peak load and the corresponding delection o beams C1 and C2, which were strengthened with one layer and two layers o carbon FRP, respectively, are 1,26.3kN (271.4 kips) and 3.6mm (1.21 inch) and 1,432.kN (322.2 kips) and 41.8mm (1.66 inch), respectively. The load-delection curves o the beams in the G and GS-series showed similar behavior like the beams in the C-series. The shear strengths and the delections corresponding to the peak loads o all 1 beams are given in detail in Table 3. Figs. 7(a) and (b) show the maximum load and delection versus the FRP ratio multiplied by its modulus o elasticity ρ E, respectively. The maximum load and delection corresponding to the maximum load o all beams except beam C2 increase almost linearly with the increase o ρ E, regardless o iber materials (carbon or glass) and coniguration (continuous sheets or strips), while those o C2 do not because this beam ailed in shear ater lexural yielding.

18 5.2. Strain distributions o the FRP, the steel stirrups, and the LVDTs The strain distributions o the FRP composites and steel stirrups just prior to shear ailures o some beams (C1, G1, GS1, and GS5), which are considered to represent typical strain distributions o all specimens, are shown in Fig. 8. The igures also include the vertical strains measured by the LVDTs attached to the beams. Because the strain gages can measure only one point strain o stirrup or FRP sheet, the eective strains o FRP calculated by Eq.(1) were also compared with the average strains measured rom LVDTs. In Fig. 8, the x-axis represents the location along the length o the beams (the distance to the section rom the let support), while the y-axis represents the measured values o the strain gages or LVDTs. Almost all o the steel stirrups reached the yield strain beore shear ailure occurred, while the maximum strains o the FRP composites just prior to shear ailures remained well below the rupture strains. The strains measured by the LVDTs were similar to those by the strain gages attached on the FRP. Fig. 9 shows the eective strain at shear strength versus the FRP ratio multiplied by its modulus o elasticity ρ E. The strain in the transverse direction at the peak load tended to decrease as the amount o the FRP increased and as the spacing o the FRP strips decreased. The strains in the transverse direction o all beams at the peak loads are presented in Table 3.

19 6. Comparison between the experimental and calculated results 6.1. Eective strain o the FRP Fig. 1 shows the experimental and calculated eective strains o the FRP or the 9 test beams strengthened with FRP. The beam C2 that ailed in shear ater lexural yielding was excluded in the comparison. The open circles, in these igures, are the eective strains o the FRP measured using the LVDTs, while the open squares are the eective strains measured using the strain gages. The eective strains ( ε e ana ) calculated according to Eq. (1) are also provided in the igure or comparison. The eective strains o FRP calculated by Eq.(1) were compared with the strains measured rom both strain gages and LVDTs. The eective strain o the FRP decreases as the amount o FRP ( ρ r ) increases, and the eective strain predicted by the considered equation shows good agreement with the experimental results. The eective strain o the FRP ( ε ), the eective strength o concrete ( v ' ), and the longitudinal e ana c strain ( ε l ), o the 1 test beams calculated by the Eqs. (7), (9), and (1) are listed in Table 5. As the amount o FRP ( ρ r ) increases, the eective strain o the FRP decreases. Accordingly, the eective compressive strength o concrete and the longitudinal axial strain o the beam are correspondingly reduced due to the decrease in ε e. To urther veriy the applicability o the proposed method in predicting the eective strain o FRP, the predicted values o the proposed method were compared with the experimental results o FRP strengthened reinorced concrete beams [17, 19-22] reported in the technical

20 literature. Although there are abundant test results o load vs. delection curves or RC beams strengthened with FRP, available test results o the eective strain o the FRP in RC beams, which ailed due to crushing o concrete or FRP rupture without debonding (or premature ailure/ peeling o), are limited. Thus, a total o 29 RC beams strengthened by FRP wrapping were compared in this study. All o the beams ailed in shear due to crushing o the concrete or FRP rupture without debonding. The maximum compressive strength o concrete o the 29 beams was 4.5 MPa. All o the beams had a rectangular cross-section. It is reminded that the proposed method is capable o predicting the eective strain o the FRP in the RC beams that ail due to crushing o the concrete or FRP rupture without debonding (or premature ailure/ peeling o). Thus the test results o the RC beams ailed due to debonding were omitted in the comparison. The comparisons between the experimental and predicted results o the eective strains o the 29 beams are shown in Fig. 11 and summarized in Table 6. Fig. 11 compares the observed and predicted eective strains o the FRP. The predicted strains were calculated by Eq.(1). The mean and variance values o the eective strain ratio ε e test / ε e ana o the 29 RC beams as predicted by the proposed method are 1. and 16.2 %, respectively. The mean and variance values o the eective strain ratio ε e test / ε e ana o the 2 RC beams without the current beam tests were.94 and %, respectively. The calculated eective strains by the existing equations in Table 1 were also compared with the observed ones as listed in Table 6. The variance values o ε e test / ε e ana as predicted by these equations were about 27.5%.

21 6.2. Shear strength o beams Twenty two more RC beams [18, 23 & 24] reported in the technical literature were added to the 29 beams in Table 7 to check the applicability o the proposed equation based on the compatibility aided truss model in predicting the shear strength o the RC beams strengthened with FRP composites. Thus, total o 51 RC beams were compared with some o the existing models and the proposed equation. Four models are considered in this comparison: Chaallal et al. s model [7], Triantaillou and Antonopoulos model [8], Khalia et al. s model [9], and Chen and Teng s model [1]. These our models are chosen because, in authors opinion, they represent a wide spectrum o prediction models available in the literatures. All o the beams were reported to ail in shear without debonding (or premature ailure/ peeling o) beore lexural yielding. The amount o the FRP composites ( ρ r ) varied rom.33 MPa to MPa. Three type o iber materials (carbon, glass, and aramid) and two types o coniguration (continuous sheets and strips) were used. To deine the shear strength o RC beams, the eective strain o the FRP, ε e, were calculated rom the equations in Table 1 and Eq.(1). This will then allow the shear resistance contributions o FRP, V, to be determined using Eq. (2). Eq. (1) is then used to calculate the shear strength o RC beams, V n. The shear strength, V n, in Eq.(1) is inluenced by not only

22 V but also V c and V s. The shear equations o V c and V s in the design code (ACI design code) were selected to calculate Eq.(1). No saety actors were considered in all shear strength calculations by the ACI code. The angle o inclination o diagonal compressive stresses to the longitudinal axis o the member (θ ) was assumed to be 45 degree in the calculation o V s and V. The comparisons between the calculated and observed shear strength o the 51 beams are shown in Fig. 12 and summarized in Table 7. Fig. 12 indicates the prediction results o each analytical model or varying ρ / ' on r c the shear strengths o the 51 RC beams. As can be seen rom Fig. 12(a), the analytical result o Chaallal et al. s model or the shear strength o 51 RC beams slightly overestimates with a mean V test /Vana. value o.93 and a COV (Coeicient o Variance) o 26.6%. As illustrated by Figs. 12 (b) and (c), the Triantaillou and Antonopoulos model and Khalia et al. s model underestimated the observed shear strength o the 51 RC beams strengthened with FRP composites. The mean values o the shear strength ratios ( V test /V ana. ) o the test beams calculated using Triantaillou and Antonopoulos model and Khalia et al. s model are 1.78 and 1.24, with coeicient o variance values o 41.2% and 32.%, respectively. As shown in Fig. 12 (d), Chen and Teng s model underestimated the observed shear strength o the 51 RC beams, which derived the eective stress o FRP by using the stress distribution actor, D FRP (ratio o the average strain to the maximum strain within the eective FRP height). It is noted that considering the simplicity o Chen and Teng s model,

23 this model predicted the shear strengths o 51 beams strengthened with FRP composites with reasonable agreement. Fig. 12 (e) shows that the proposed solution method based on the smeared strains provides improved results compared to those obtained rom the existing models or the dierent values o ρ / ' investigated in this research. It can be also seen rom Fig. 1 (e), the numerical r c comparison o the proposed method indicates reasonable correlations with the experimental shear strength o the 51 RC beams with a mean o 1.4 and a COV o 2.6%. It is, thereore, worth noting that the compatibility aided truss model based on the average strains adopted in this study to calculate the eective strain o FRP is reasonably applicable to calculate the shear strengths o the RC beams strengthened with FRP composites. However, the validity o the proposed model needs to be veriied against more independent experimental data, because the shear equations o V c and V s in the design code (ACI design code) used to calculate Eq.(1) may not predict the exact shear resistance contributions o concrete and steel stirrups. Hence, the comparisons reported here are limited to those obtained rom the current study. More elaborate comparisons with the test results o V are needed or uture research. 7. Conclusions The eective strain o the FRP plays an important role in predicting the shear strength o RC beams strengthened with FRP composites. In this paper, the eective strains o the FRP

24 used as strengthening material in RC beams were studied. Ten RC beams strengthened by FRP were tested and the strain o the FRP composites was measured at speciied load intervals by the LVDTs attached to the beams and electrical strain gages installed on the suraces o the FRP. The test results indicated that the eective strain o the FRP at shear ailure decreased as the amount o FRP increased and also as the spacing o the FRP strips decreased. An equation based on the compatibility-aided truss model was proposed to predict the eective strain o the FRP. The eective strain o the FRP calculated by the proposed equation depended on the interaction between the amount o FRP and the compressive strength o concrete. Even i the proposed Eq. (1a) was rather tedious to calculate the eective strain, this equation will help to understand the eects o the eective strength o concrete and the stress conditions o the materials o the beams at shear ailure on the eective strain o the FRP. The obtained results showed that the proposed equation predicted the eective strains o the 29 FRP-strengthened RC beams with reasonable agreement. In addition, the numerical comparison o the proposed method indicates reasonable correlations with the experimental shear strength o the 51 RC beams with a mean o 1.4 and a COV o 2.6%. The proposed method was capable o predicting the eective strain o the FRP in the RC beams that ail due to crushing o the concrete or FRP rupture without debonding. Thus

25 urther rigorous research containing the eect o bond ailure on the eective strain o the FRP composites is needed. Reerences [1] Ceroni F, Pecce, M, Matthys S, Taerwe L. Debonding strength and anchorage devices or reinorced concrete elements strengthened with FRP sheets. Compos Part B-Eng 28;39(3): [2] Teng JG, Chen GM, Chen JF, Rosenboom OA, Lam L. Behavior o RC Beams Shear Strengthened with Bonded or Unbonded FRP Wraps. J Compos Constr, ASCE, 29;15(5): [3] Deialla A, Ghobarah A. Strengthening RC T-Beams Subjected to Combined Torsion and Shear Using FRP Fabrics: Experimental Study. J Compos Constr, ASCE 29;15(3): [4] Grande E, Imbimb M, Rasulo A. Eect o Transverse Steel on the Response o RC Beams Strengthened in Shear by FRP: Experimental Study. J Compos Constr, ASCE 29;15(5): [5] Fico R, Prota A, Manredi G. Assessment o Eurocode-like design equations or the shear capacity o FRP RC members. Compos Part B-Eng 28;39(5): [6] Li A, Diagana C, Delmas Y. Shear strengthening eect by bonded composite abrics on RC beams. Compos Part B-Eng 22;33(3): [7] Chaallal O, Nollet MJ, Perration D. Strengthening o Reinorced Concrete Beams with Externally Bonded Fibre-Reinorced-Plastic Plates. Canadian J Civil Eng 1998;25(4): [8] Triantaillou TC, Antonopoulos C. Design o Concrete Flexural Members Strengthened in Shear with FRP. J Compos Constr, ASCE, 2;4(4): [9] Khalia A, Gold WJ, Nanni A, Aziz A. Contribution o Externally Bonded FRP to Shear Capacity o RC Flexural Members. J Compos Constr, ASCE 1998;2(4): [1] Chen JF, Teng JG. Shear Capacity o Fiber-Reinorced Polymer-Strengthened Reinorced Concrete Beams: Fiber Reinorced Polymer Rupture, J Struct Eng 23; 129(5): [11] Lee JY, Kim SW. Torsional Strength o Reinorced Concrete Beams Considering Tension Stiening Eect, J Struct Eng, ASCE 21;136(11)(to be issued). [12] Vecchio F, Collins MP. The Modiied Compression-Field Theory or RC Elements

26 Subjected to Shear. ACI Struct J 1986; 83(2): [13] Lee JY, and Watanabe F. Shear Design o RC Beams with Shear Reinorcement Considering Failure Modes. ACI Struct J 2; 97(3): [14] Mohamad YM, Lee JY, Hsu TTC. Cyclic Stress-Strain Curves o Concrete and Steel Bars in Membrane Elements. J Struct Eng 21;127(12): [15] Vecchio FJ, Collins MP. Predicting the Response o RC Beams Subjected to Shear Using the Modiied Compression Field Theory. ACI Struct J 1988;85(4): [16] Watanabe F, Lee JY. Theoretical Prediction o Shear Strength and Failure Mode o Reinorced Concrete Beams. ACI Struct J1998; 95(6): [17] Ohuchi H, Ohno S, Katsumata H. Seismic strengthening design technique or existing bridge columns with CFRP. Proceedings o the 2nd International Workshop on Seismic Design and Retroitting o Reinorced Concrete Bridges, R. Park, Ed., 1994: , Queenstown, New Zealand. [18] Araki N, Matsuzaki Y, Nakano K, Kataoka T. Experimental Study on Shear Capacity o RC Beams using Sheet Type Fiber. Proceedings o the JCI 1997;19(2): [19] Uji K. Improving the shear capacity o existing reinorced concrete members by applying carbon iber sheets. Trans. Japan concrete Institute 1992;14: [2] Miyauchi K, Inoue S, Nishibayashi S, Tanaka Y. Shear behavior o RC beam strengthened with CFRP sheet. Trans. Japan concrete Institute 1997;19: [21] Funakawa I, Shimono K, Watanabe T, Asada S, Ushijima S. Experimental study on shear strengthening with continuous iber reinorcement sheet and methyl methacrylate resin. Non-Merallic (FRP) Reinorcement or Concrete Structures, Proceeding o the Third International Symposium, Sapporo, Japan,1997. p [22] Umezu K, Fujita M, Nakai H, Tamaki K. Shear behavior o RC beams with aramid iber sheet. Non-Merallic (FRP) Reinorcement or Concrete Structures, Proceeding o the Third International Symposium, Sapporo, Japan, p [23] Kato H, Kojima T, Takagi N, Hamada Y. Experimental Study on Shear Reinorcement o RC Beams by using Carbon Fiber Sheets. Proceedings o the JCI 1996:18(2): [24] Ishizaki K, Maruyama Y, Nakano K, Kataoka T. Size Eect in Shear Strength o o RC Beams Retroitted with CFRP Sheet. Proceedings o the JCI 1997;19(2):

27 Table Table 1 Various equations or the eective strain o FRP. Reerence e Note Chaallal et al. 7).8 u - Triantaillou and Antonopoulos 8) 2/3.3 ( c ').17 E 2/3.47 ( c ').48 E u u For ully wrapped CFRP (CFRP racture) For ully wrapped AFRP (AFRP racture) Khalia et al. 9) (.5622( E ) E.778).5 - u u Chen and (1 ) / 2 Teng 1) where zt / zb, zt dfrp, t, zb.9d For FRP rupture ailure

28 Table 2 Material properties and calculated results o analyzed three beams. Beams c ' (MPa) Shear steel reinorcement t (%) ty (MPa) (Deg) Longitudinal tensile reinorcement l (%) ly (MPa) r (MPa) FRP composites r (%) E (GPa) (%) Balanced FRP ratio by Eq. (4) b (%) Eective strain e (%) Failure mode B , Balanced ailure B , Mode 1 B , Mode 1 (Note: 1 MPa = 145 psi)

29 Table 3 Speciication o specimens and test results. FRP composites Vertical strains (%) Serie s Beams Fiber Distrib ution s (mm) n (%) E (MPa) P n (kn) n (mm) F. M. e LVDT e rp se steel C G GS Note: C1 Carbon sheets C2 Carbon sheets S G1 Glass sheets F G2 Glass sheets S G3 Glass sheets S G4 Glass sheets S GS1 Glass strips S GS1a Glass strips S GS3 Glass strips S GS5 Glass strips S (Note: 1 MPa = 145 psi; ; 1 kn =.225 kip; 1mm =.394 in.) s is the spacing o FRP strips, n is the number o FRP layers; P is the maximum load; n is the ratio o FRP, F.M.: ailure mode, is the midspan delection corresponding to n P ; n e LVDT is the strain measured using the are the strains measured using the strain gage attached on the FRP and the steel vertical LVDT; e rp and e steel stirrups, respectively; F.M.: ailure modes; S : shear aiure due to concrete crushing beore yielding o the lexural steel; F : shear ailure ater yielding o the lexural steel. Table 4 Material properties o steel rebars and FRPs. Materials Section area t y or r y or r E s or E (mm 2 ) (mm) (MPa) (MPa) D1 (Stirrups) , D25 (Long. Reinorcement) , Carbon FRP (Shear reinorcement) ,51 25, Glass FRP (Shear reinorcement) , t is the thickness o FRP, r and y are the rupture strain o FRP and yield strain o steel bars, respectively, r and y are the rupture strength o FRP and yield strength o FRP bars, respectively, E and Es are the modulus o elasticity o FRP and steel bars, respectively. (Note: 1 MPa = 145 psi; 1mm =.394 in.)

30 Table 5 Experimental and calculated results o the 1 test beams. Series Beams C G GS ' c r v c ' (MPa) Analytical results l e ana Test results e LVDT e rp e LVDT e ana e rp e ana C C * 1.289* G G G G GS GS1a GS GS Ave V. C % 8.47 % e ana is the eective stain o the FRP calculated by the proposed method; Ave. is the mean value; V.C is the variance coeicient. * C2 ailed in shear ater lexural yielding. (Note: 1 MPa = 145 psi) Table 6 Experimental and calculated eective strains o FRP in RC beams. Beams e test (%) *1 *2 *3 *4 *5 Beams e test (%) *1 *2 *3 *4 *5 3 19) AN-1/5 Z-3 2) BS12 17) AN-1/2 Z-3 2) BS24 17) CN-1/L Z-2 2) BM6 17) S-2 21) BM12 17) S-4 21) BM18 17) CS2 22) BM24 17) C BL6 17) G BL12 17) G BMW6 17) G BMW12 17) G BMW24 17) GS ) GS1a ) GS GS Ave V. C.(%) *1= e test / 7) e chaallal, *2= e test / 8) e Tri.& Anto, *3= e test / 9) e Khalia, *4= e test / 1) e Chen& Teng, *5= e test / e Eq.(1)

31 Table 7 Experimental and calculated shear strengths o RC beams. Beams Vn exp. (kn) *1 *2 *3 *4 *5 Beams Vn exp. (kn) *1 *2 *3 *4 *5 3 19) N-E1Z*2 23) BS12 17) N-E1*2 23) BS24 17) N-E1S 23) BM6 17) N1-E1S 23) BM12 17) RB-CF-45 18) BM18 17) RB-CF-64 18)) BM24 17) RB-CF-97 18)) BL6 17) RB-CF )) BL12 17) RB-CF )) BMW6 17) RB-AF-6 18)) BMW12 17) RB-AF-9 18)) BMW24 17) RB-AF-12 18)) ) No.2 24) ) No.3 24) AN-1/5 Z-3 2) No.7 24) AN-1/2 Z-3 2) No.8 24) CN-1/L Z-2 2) C S-2 21) G S-4 21) G N-E1Z 23) G N-E1 23) G N1-E1 23) GS N-E2 23) GS1a N1-E2 23) GS N-H1 23) GS N1-H1 23) Ave V. C.(%) *1= Vn exp. / Vn chaallal, *2= Vn exp. / Vn Tri.& Anto, *3= Vn exp. / Vn Khalia, *4= Vn exp. / Vn Chen& Teng, *5= Vn exp. / Vn Eq.(1)

32 Figure V cotθ 2 2 d 1 v A θ α V Fig. 1 A truss model o a FRP strengthened RC beam. cot 2 θ Failure mode 1 Concrete crushing beore FRP rupture Failure mode 2 FRP rupture FRP, rp Steel, tv Shear strain Principal compressive stress o concrete, 2 ' c Eective compressive strength o concrete, v c ' Fig. 2. Stresses o materials versus strain o RC beams strengthened with FRP composites.

33 v lt 2 = v c ' c (, v ) l lt 2θ v 1 = v ρ l l γ lt 2 γ ε 2 = γ lt ( εl, ) 2 2θ ε 1 ε c (, v ) t lt ρ = ρ + ρ tv tv + ρ tv tvy r γ lt ( εt, ) 2 γ lt = ( ε r, ) 2 Fig. 3. Mohr's stress and strain circles at the boundary between ailure mode 1 and ailure mode 2. FRP reinorcement, ρ (MPa) p by right term o Eq.(1a) p by let term o Eq.(1a) '= 6MPa c Failre mode 1 Failre mode 2 '= 4MPa c ρ =.16 '= 2MPa c ρ =.8 C ρ =.244 B A Strain o FRP composites, ε Fig. 4. Calculation o the eective strain o FRP composites at shear ailure. (Note: 1 MPa = 145 psi)

34 Fig. 5. Dimensions and reinorcement o test beams o GS-series. (Note: 1mm =.394 in.) Load (kn) G-series Load (kn) G-series 3 C1 C Delection (a) C-series (mm) 3 G1 G2 G3 G (b) G-series Delection (mm) Load (kn) GS-series GS1 GS1a GS3 GS Delection (mm) (c) GS-series Fig. 6. Load versus delection curves o the test beams.(note: 1 kn =.225 kip; 1mm =.394 in.)

35 Maximum load (kn) Carbon-sheets Glass-sheets Glass-strips ρ E (GPa) C2 Delection correspoing to the maximum load (mm) Carbon-sheets Glass-sheets Glass-strips ρ E (GPa) C2 (a) Maximum load (b) Delection Fig. 7. Maximum load and delection o the test beams. (Note: 1 GPa = 145 ksi; 1mm =.394 in.).25.2 C1 Rupture strain o GFRP LVDT rp-gauge steel-gauge.25.2 C2 Rupture strain o GFRP LVDT rp-gauge steel-gauge.25.2 G1 Rupture strain o GFRP LVDT rp-gauge steel-gauge.15 Rupture strain o CFRP.15 Rupture strain o CFRP.15 Rupture strain o CFRP Yield strain o steel.5 Yield strain o steel Yield strain o steel G2 Rupture strain o GFRP LVDT rp-gauge steel-gauge.25.2 G3 Rupture strain o GFRP LVDT rp-gauge steel-gauge.25.2 G4 Rupture strain o GFRP LVDT rp-gauge steel-gauge.15 Rupture strain o CFRP.15 Rupture strain o CFRP.15 Rupture strain o CFRP Yield strain o steel.5 Yield strain o steel Yield strain o steel GS1 Rupture strain o GFRP LVDT rp-gauge steel-gauge.25.2 GS1a Rupture strain o GFRP LVDT rp-gauge steel-gauge.25.2 GS3 Rupture strain o GFRP LVDT rp-gauge steel-gauge.15 Rupture strain o CFRP.15 Rupture strain o CFRP.15 Rupture strain o CFRP Yield strain o steel.5 Yield strain o steel Yield strain o steel GS5 Rupture strain o GFRP Rupture strain o CFRP LVDT rp-gauge steel-gauge.1.5 Yield strain o steel

36 Fig. 8. Strain distributions o the FRP, the steel stirrups, and the LVDTs. (Note: 1mm =.394 in.) ε e / ε r ε (carbon-sheets) e-lvdt ε (carbon-sheets) e-gauge ε (glass-sheets) e-lvdt ε (glass-sheets) e-gauge ε (glass-strips) e-lvdt ε (glass-strips) e-gauge ρ E (GPa) Fig. 9. Eective strain o the test beams at shear strength. (Note: 1 GPa = 145 ksi) Eective strain o FRP at shear ailure(ε e ) Rupture strain o GFRP Rupture strain o CFRP ε e-ana ε e-lvdt ε e-gauge ρ r / ' c

37 Fig. 1. Comparison between the experimental and calculated eective strains o FRP in 9 test beams. 1.5 ε e-test / ε e-ana Mean : 1. V. C. : 16.2 % CFRP-existing data GFRP-current test CFRP-current test ρ r / c ' Fig. 11. Comparison between the experimental and calculated eective strains o FRP in RC beams reported in the technical literature.

38 V test / V ana (a) Chaallal et al. Mean :.93 V. C. : 26.6 % AFRP CFRP GFRP V test / V ana (b) Triantaillou & Antonopoulos AFRP CFRP GFRP ρ / ' r c ρ / ' r c Mean : 1.78 V. C. : 41.2% V test / V ana (c) Khalia et al. Mean : 1.24 V. C. : 32.% AFRP CFRP GFRP V test / V ana (d) Chen & Teng Mean : 1.28 V. C. : 26.9% AFRP CFRP GFRP ρ / ' r c ρ / ' r c V test / V ana (e) Proposed equation Mean : 1.4 V. C. : 2.6 % AFRP CFRP GFRP ρ / ' r c Fig. 12. Comparison between the experimental and calculated shear strengths RC beams strengthened with FRP composites reported in the technical literature.