Failure Diagrams of FRP Strengthened RC Beams

Transcription

1 Failure Diagrams o FRP Strengthened RC Beams Abstract Bo GAO a, Christopher K. Y. LEUNG b and Jang-Kyo KIM a* a Department o Mechanical Engineering and b Department o Civil Engineering Hong Kong University o Science & Technology Clear Water Bay, Hong Kong, China This is the Pre-Published Version Amongst various methods developed or strengthening and rehabilitation o reinorced concrete (RC) beams, eternal bonding o ibre reinorced plastic (FRP) strips to the beam has been widely accepted as an eective and convenient method. The eperimental research on FRP strengthened RC beams has shown ive most common modes, including (i) rupture o FRP strips; (ii) compression ailure ater yielding o steel; (iii) compression ailure beore yielding o steel; (iv) delamination o FRP strips due to crack; and (v) concrete cover separation. In this paper, a ailure diagram is established to show the relationship and the transer tendency among dierent ailure modes or RC beams strengthened with FRP strips, and how ailure modes change with FRP thickness and the distance rom the end o FRP strips to the support. The idea behind the ailure diagram is that the ailure mode associated with the lowest strain in FRP or concrete by comparison is mostly likely to occur. The predictions based on the present ailure diagram are compared to 33 eperimental data rom the literature and good agreement on ailure mode and ultimate load has been obtained. Some discussion and recommendation or practical design are given. Keywords: Failure mode; RC beam; Strengthen; Diagram; Fibre reinorced plastic * Corresponding author. Tel: ; Fa: ; mejkkim@ust.hk 1

2 1. Introduction Inrastructure repair and rehabilitation has become an increasingly important challenge to the concrete industry in recent years. Upgrading structural load capacity is a substantial part o the rehabilitation market, and seismic retroit o concrete components in earthquake regions is now becoming a mainstream. As a combined result o structural rehabilitation needs, strengthening and rehabilitation o concrete structures have become the industry s major growth area. Amongst various methods developed or strengthening and rehabilitation o reinorced concrete (RC) beam structures, eternal bonding o ibre reinorced plastic (FRP) strips to the beam has been widely accepted as an eective and convenient method. The main advantages o FRP include high strength and stiness, high resistance to corrosion and chemicals, as well as light weight due to low density. The retroitting can be applied economically, as there is no need or mechanical iing and surace preparation. Moreover, the strengthening system can be easily maintained. Signiicant progress has been made based on eperiments, theoretical analysis and numerical simulation to demonstrate that the bonding o FRP strips to the tension soit o reinorced concrete beams can improve much the ultimate leural strength and stiness, although some reduction in ductility o the beam is caused. In strengthening reinorced concrete beams with FRP strips, dierent ailure modes have been observed [1-3]. Generally speaking, there eist si distinct ailure modes (see in Fig. 1), as described in the ollowing: (i) Compression ailure beore yielding o steel: the concrete crushes in compression (i.e. the strain in the concrete eceeds the ultimate value o ) beore yielding o reinorcing steel and racture o FRP strips; (ii) Compression ailure ater yielding o steel: the reinorcing steel yields due to tensile leure. This is ollowed by crushing o the concrete in the compression zone, beore the tensile rupture o the FRP strips; 2

3 (iii) Rupture o FRP strips: the FRP strips rupture at the ultimate strain ollowing the yielding o reinorcing steel rebar in tension; (iv) Shear ailure: the shear cracks etend rom the vicinity o the support to the loading point, when the shear capacity o the beam is eceeded; (v) Delamination o FRP strips: delamination o CFRP strip occurs rather catastrophically in an unstable manner, with a thin layer o concrete residue attached to the delaminated FRP sheets. The crack initiates rom the end o FRP strips or the bottom o a leural or shear/leural crack in the concrete member; (vi) Concrete cover separation: ater crack initiation at the CFRP strip end, the CFRP strip is gradually peeled o with lumps o concrete detached rom the longitudinal steel rebar. These modes can be divided into two general categories, namely leural ailures and local ailures. The leural ailures include compression ailure beore yielding o steel, compression ailure ater yielding o steel and rupture o FRP strips; Shear ailure, delamination o FRP strips and concrete cover separation belong to local ailures. Fleural ailure modes are a typical o those encountered in conventional concrete beams, and, thereore, the perception on ailure mechanism and analytical methods or these ailure modes have already been successully established. Although FRP rupture without yielding o steel reinorcement is sometimes regarded as a kind o leural ailure mode, it is unlikely to occur unless the steel in tension is located very near the centre o beam. In most leural equations in the literatures or design recommendations, the most preerred ailure mode to be designed or is compression ailure ollowing yielding o steel reinorcement. Rupture o FRP strips ollowing yielding o steel reinorcement is also acceptable. In comparison, compression ailure beore yielding o steel should be avoided as ar as possible. In the above, the steel reinorcement mostly reers to steel rebar in tension. The yielding o tension steel rebar can ensure the ormation o large leural cracks, which provides warning beore ultimate ailure. 3

4 Shear ailure is caused generally by low shear reinorcement due to relatively large stirrup spacing. It may also occur when only leural strengthening is applied, because the FRP strips along the bottom o reinorced concrete beams does not improve the shear strength o beam remarkably. It is ound out that, however, restoring or upgrading beam shear strength using side FRP strips can result in increased shear strength and stiness by substantially reducing shear cracking [4-6]. Many parameters including reinorcement coniguration (U strip, side strip, ull wrap), FRP orientation, the use o mechanical type anchors, concrete strength, steel shear reinorcement and shear span to depth ratio [7-9], have been studied. Generally speaking, shear ailure can be eliminated by the appropriate shear strengthening o the beam as mentioned above, and it is not to be discussed in the ollowing sections. In the delamination o FRP strips, the bond between the FRP strip and the concrete ails in a sudden manner as a result o the catastrophic propagation o a crack along the FRP concrete interace. In general, several reasons may cause this ailure, such as: (a) technical laws including imperections in the spreading o the adhesive and signiicantly uneven concrete tensile aces; (b) leural and leural/shear cracks in the concrete that result in horizontal interace cracks developed rom the bottom tip o the leural cracks; and (c) high shear and normal stress concentration at the end o FRP due to discontinuity [10]. Correct preparation and operation can avoid aorementioned technical laws. To analyse the initiation o ailure at the end o FRP strips, a number o models are available. These include closedorm high order analytical models to solve or stress distributions [11,12], shear-capacitybased models [13,14], and interacial stress-based models [15-17]. However, eperimental results show that, delamination along the concrete/frp interace is most likely to occur rom leural and leural/shear cracks. High stress concentration at the end o FRP strips may induce concrete cover separation instead o delamination. Thereore, only delamination resulting rom the leural and leural/shear cracks on the tensile side is considered in the ollowing, and the eisting analytical models will be discussed. 4

5 Concrete cover separation is a very common ailure mechanism observed in eperimental work. For this ailure mode, a crack initiates in the vicinity o one o the FRP plate ends, then develops to the level o the tension steel reinorcement, and propagates horizontally towards the mid span along the steel rebar. It is noticed that in the process many shear/leural cracks are developed in the concrete cover orming tooths between the cracks. Based on this mechanism, many theoretical models have been built. From the design point o view, the relationship and the transition guideline among the various ailure modes have to be understood. Currently there are very ew papers that study the varying trend o ailure mode in terms o the change o strengthening parameters (e.g. FRP thickness, FRP length, etc.), and identiy which ailure mechanism is dominant or the beam design. The objective o this paper is to build a diagram showing the relationship and the transition among dierent ailure modes or RC beams strengthened with FRP strips, and how ailure modes vary with FRP thickness and the distance rom the end o FRP strips to the support. The ailure mode prediction diagram is useul in establishing an FRP material selection procedure or eternal strengthening o RC beams. A review o previous theoretical models or these ailure modes is given irst, and appropriate epressions are chosen or ailure mode prediction. A step-by-step procedure to establish the ailure mode diagram is also presented. Furthermore, a design eample is provided to demonstrate the applicability o this approach. The applicability o the approach will then be veriied with a signiicant number o eperimental results. Finally, some discussions and recommendations or practical design are given. 2. Theoretical epressions or various ailure modes 2.1 Fleural ailure modes 5

6 To date, numerous leural design equations have been produced, and also eisting research suggests that the ultimate leural strength o FRP strengthened RC beams can be predicted using eisting RC beam design approaches with appropriate modiications to account or the brittle nature o FRPs [2,10,16,18-23]. Some similar assumptions in leural strength design equations are (a) plane section remaining plane ater bending; (b) zero tensile strength in concrete; (c) adhesive being omitted; and (d) the perect bonding between the concrete and FRP plate. Fig. 2. shows the cross section o a rectangular beam subjected to bending and the resultant strain distribution along the depth o the beam as well as a simpliied equivalent rectangular stress block. Notice that d, d, and d denote the depths o compressive steel, tensile steel and FRP strips, respectively; As and As are the area o tensile and compressive steel reinorcement; b c and b are the width o concrete and FRP strips; and, h, and h are the depth o the neutral ais, concrete beam, and concrete cover, respectively. In addition, ε c, ε s, ε s, andε are the strains o concrete, tensile steel rebar, compressive steel rebar and FRP strips, respectively. With the reerence to Fig. 2., the internal orce components related to concrete and FRP strips are, C c = α1 c b c β1 (1) T = E ε A (2) where α 1 (the ratio o the uniorm stress in the rectangular compression block to the maimum compressive strength) and β 1 (the ratio o the depth o the rectangular compression block to the depth to the neutral ais). Dierent values o α 1 and β 1 are deined as ollows. El-Mihilmy and Tedesco [2] set α 1 and β 1 to be 0.85 and c, respectively. In Ng and Lee [23], the adopted values are 0.67 and 0.9 or α 1 and β 1. Considering the eect o compressive concrete strength on these two actors, Chaallal et al. [19] deined α 1 and β 1 as ollows, 6

7 which are also recommended in this paper. α = c 0.6 (3) β = c 0.6 (4) Since there are three main leure ailures, two balanced limited values o cross section area o FRP are employed, A,min and A,ma. I A < A,min, the rupture o FRP strips mode can dominate. I A,min < A < A,ma compression ailure ater yielding o steel must take place. I A >A,ma, compression ailure beore yielding o steel is to occur. In the calculation o A,min, ε c = ε cu (0.0035) and ε = ε u (the racture strain o FRP) are assumed to happen simultaneously. As the ailure mode transitions rom FRP rupture to compression ailure, dierent epressions or A,min can be obtained or dierent compressive steel conditions, hεcu = (5) ε + ε cu u A α1 c bc β1 + Esε s As y As = ε d min, E s = < ε ε sy (6) u A α1 c bc β1 + y As y As = ε d min, E s = ε ε sy (7) u In the calculation o A,ma, ε c = ε cu (0.0035) and ε s = ε sy (the yielding strain o tension steel) are assumed to occur simultaneously. As the ailure mode transitions rom compression ailure ater yielding o steel to compression ailure beore yielding o steel, A,ma or dierent compressive steel conditions, can be obtained as ollows. dε d cu =, ε = ε + ε cu sy (8) α1 c bc β1 + Esε s As y As A d = ε ma, E s = < ε ε sy (9) 7

8 A α1 c bc β1 + y As y As = ε d ma, E s = ε ε sy (10) Although the main objective o this ailure diagram is to show the relationship and the transition among dierent ailure modes, it can also predict the ultimate leural strength o FRP strengthened RC beams. Only brie descriptions or epression are presented in Appendi A. 2.2 Delamination o FRP strips Besides the end o FRP strips, leural and leural /shear cracks are also possible locations or delamination to occur. While the beam is loaded, these cracks tend to open and may induce high interacial shear stress, thus resulting in crack propagation along the interace. Compared to the eisting stress analysis or delamination rom the end o FRP, not much research has been carried out or delamination initiating rom cracks. Triantaillou and Plevris [10] suggested that the ailure was due to vertical (v) and horizontal (w) concrete crack openings, which were resulted rom the dowel action and aggregate interlock mechanisms. Also, it was assumed that the dowel deormation in the longitudinal steel and the FRP at the crack location were primarily due to shear. Thereore, when the shear orce reached a critical value, the ailure occurred as ollows, ( t ) V v cr = G s A s + G b w (11) cr With the equation, the corresponding load capacity could be obtained. Nevertheless, (v/w) cr that was a characteristic property o the FRP concrete bond, was not supported by necessary eperimental results. In the study by Buyukozturk and Hearing [1], it was shown that leural cracks in large moment region could initiate interacial racture in shear mode, and leural/shear cracks in mied shear and moment region could induce mied mode racture. With the 8

9 concept o racture mechanics, when the strain energy release rate reaches the interacial racture resistance, ailure takes place. The critical strain energy release rate can be measured with the single lap test. Normal and shear stress distributions along the interace between concrete and FRP have been studied in many papers. O note is that the normal stress perpendicular to the plate under leural cracks is compression due to bending. Since compressive normal stress cannot lead to delamination, only shear stress under the cracks was responsible or delamination [24]. At the two sides o a crack, the maimum shear stress τ ma at the adhesive/concrete interace can be calculated i the longitudinal stress in FRP plate is known [17]. An approimate equation or τ ma is given by [20] where the was the aial stress in the FRP plate. Gat τ ma = E t (12) A theoretical ramework was developed to analyse the delamination at the location o a leural crack in the beam [24]. A racture mechanics analysis was applied to get the relationship among M (moment), a (crack length), and w (crack mouth). The iterative calculation gave rise to M or a given crack size. Then, the maimum shear stress concentration at the crack could be obtained rom a τ wg ma = 2t. (13) a By repeating the computation or various crack sizes, the relationship between τ ma and M could be established. It was shown that crack induced delamination o FRP had much in common with debonding ailures observed in the simple shear test [25]. In the literature, several bond strength models based on the racture mechanics have been developed [26,27]. By modiying 9

10 these models and with the empirical itting o eperimental data, the ollowing epression was obtained by Teng et al [25]: c E σ, ma = 1.1β (14) t 2 b / bc β = (15) 1+ b / b c in which σ, ma was the maimum tensile stress permitted in FRP plate. 1.1 is a actor that will provide the best it to eperimental results. When the tensile stress in FRP strips reaches σ,ma in a strengthened RC beam subjected to bending, FRP debonding occurs. Obviously, with known σ, ma, the maimum moment or load capacity o the beam can be calculated. Since this model can provide reasonable prediction while being simple or practical use, it is employed in this paper. 2.3 Concrete cover separation In an eort to identiy the strength o a strengthened RC beam ailed by concrete cover separation, many studies have bean carried out and several analytical models were ormulated. In general, two categories o analytical theoretical solutions eist, including interacial stress model and tooth model. For interacial stress model, most papers attempt to predict the stress distribution along the interace between FRP and concrete, especially stress concentration at the end o FRP. A simpliied and approimate analytical model to produce the shear and normal stress concentrations at the cut o point o FRP strips was developed by Roberts [15]. Actually, this model has been widely accepted by many researchers, and equations based on its modiication were given in many studies [3, 28]. Also, the papers by Malek et al. [17] and Saadatmanesh and Malek [20] developed a methodology based on the linear elastic behaviour o the material and compatibility o deormation to predict the 10

11 interacial stresses. Moreover, other analytical models considering more inormation, such as orthotropic material properties, have been developed [29,30]. Elastic models are usually not accurate in predicting the ailure load [25]. Also, some elastic models are cumbersome and not suitable or hand calculation. In act, or concrete cover separation, an inclined concrete crack is always observed to orm at the plate end beore the ultimate loading is reached in the eperiments. This means that the elastic analysis is no longer valid when ailure is approached. On the other hand, using the concept o concrete tooth, tooth-based models have been developed [31,32]. A concrete tooth is a part o the concrete cover between two adjacent cracks. It deorms like a cantilever under the action o horizontal shear stresses at the bottom o the concrete beam. Concrete cover separation was deemed to occur when the tensile stress at the root o the tooth eceeded the tensile strength o concrete. Knowing the minimum crack spacing, the critical shear stress can be determined by using conventional cantilever beam theory, based on the above ailure criterion. Herein, the critical shear stress is assumed to act over an eective length determined rom empirical itting o eperimental data. Then, rom stress equilibrium o the FRP plate over the eective length, the limited maimum tension stress in FRP can be calculated, and thus the ultimate load or moment o the strengthened beam can be obtained. A major limitation o the approach is that the cantilever length (i.e., the concrete cover depth) is very short compared with its height (which is the minimum crack spacing). As a result, the conventional cantilever beam theory employed to obtain the relation between the tensile stress at the root o the tooth and the applied shear stress is not valid. In the ollowing, a new model is proposed to predict the ailure o the concrete tooth. This analytical epression was developed or predicting the stress concentrations in concrete near the tension rebar closest to the cut o point o the FRP strip, and then obtaining the load capacity based on a speciic ailure criterion. The ollowing assumptions were made: (i) linear elastic and isotropic behaviour or concrete, FRP, epoy, and steel reinorcement, (ii) perect 11

12 bonding between concrete and FRP strips, and (iii) linear strain distribution through the ull depth o the section with cracked concrete. The methodology is implemented in two stages: I) prediction o the tensile stresses in the FRP strips at the curtailments and corresponding shear stress at the location o steel bar in tension assuming ull composite action; and II) solving the stress concentrations caused by reverse tensile orce o FRP strips at the curtailment location due to the cut o o FRP strips, and comparing the superposed stresses with the concrete strength. In the second stage, the inite element method (FEM) is employed to obtain accurate stress proiles in the model, and a statistical analysis o eperimental results gives rise to a modiication actor that will lead to accurate predictions. In the irst stage, i considering the ull composite action and elastic behaviour, the tensile stress o FRP strips at the curtailment location, 0, can be obtained rom conventional beam theory as M = 0 0 ( h ). (16) I Herein, I is the cracked transormed moment o inertia o beam cross section in terms o the FRP plate, and M 0 is the bending moment at the plate curtailment location. The shear stress in concrete near the tension rebar closest to the cut o point o FRP strip, τ I 0, is V τ I 0 = 0 ( h ) b t. (17) Ib c where V 0 is the shear orce at the plate curtailment location. In the second stage, since the aial stress 0 at the end o FRP does not actually eist, an opposite orce, - Ψ 0b t ( t represents the thickness o FRP strips) is applied, to the end o FRP plate as shown in Fig. 3. As shown above, one can epect that many cracks appear in the tension side o the beam. The crack spacing model or conventional reinorced concrete is etended or calculating the minimum stabilized crack spacing, o RC beams with eternally bonded FRP plate, as presented below, l min in the case 12

13 Ae t lmin =. (18) ( u O + u b ) s bar In this equation, u s and u is the average bond strength or steel/concrete and FRP/concrete, respectively. O bar is the total perimeter o the tension bars, and A e is the area o concrete in tension. Also, one can take us = cu and u = cu. Indeed, the results are ound not too sensitive to the eact value chosen or the parameter u. In this model, Ψ is an empirical unction obtained rom empirical itting o eperimental results. It is ound that a complete quadratic equation o Ψ in terms o L s / L and b / bc, as given in Eqn (19) below, will give the best agreement with test results. Ψ = b ( L s / L) ( b / b c ) ( L s / L) ( / ) ( / L) ( / ), / L 0. 1 b c L s b b c L s 2 (19) in which, L -s and L represent the distance rom the end o FRP to the support and total span length, respectively. The comparison between predicted and eperimental values or 39 strengthened beams, with and without the modiication actor Ψ, are shown in Fig. 4. The details o selected samples are presented in Gao et al. [33]. From the igure, it is clear the modiication actor is necessary to obtain good agreement between predicted and eperimental results. Under the applied orce in Fig. 3, we assume that complete shear stress transer between FRP and concrete takes place over lmin, the concrete cover block nearest to the end o FRP strips. When the individual concrete block at the end o FRP strips is subjected to a orce ( Ψ 0 b t ), the vertical normal stress and shear stress in concrete near the tension rebar closest to the cut o point o FRP strips in stage II, σ II 0 andτ II 0, can be calculated. As mentioned, the cantilever beam length is too short compared to its depth or the conventional cantilever beam theory to be valid. Thereore, the inite element method (FEM) 13

14 is applied to obtain σ II 0 andτ II 0. The rectangular cover region (one piece o tooth) between two cracks is modelled, and a unit orce is applied at the end o FRP strips or convenience. σ II,unit 0 andτ II,unit 0, the vertical normal and shear stresses or a unit orce in stage II, can be obtained. We have attempted to solve the problem with three dierent models: (i) a 3D model with the FRP and adhesive considered (Fig. 5a), (ii) a 2D model with the FRP and adhesive considered (Fig. 5b), and (iii) a 2D model neglecting the presence o adhesive and FRP, with loading applied directly onto the concrete (Fig. 5c). The results indicate that as long as an appropriate modiication actor (obtained rom empirical itting) is used with the inite element results, each o the models can predict ailure loads in good agreement with eperimental data. For convenience, we have decided to adopt the simplest model (Fig. 5c) or urther analysis. More details on the models and comparisons with test results can be ound [33]. In practical design, it is inconvenient to run inite element analysis every time. A better alternative is to provide equations or, σ II,unit 0 andτ II,unit 0, the stresses resulted rom a unit load applied on the plate end, based on a series o inite element analysis. From the geometry o the problem, it is clear that the stresses are a unction o lmin / h, where lmin is the minimum stabilized crack spacing and h is the depth o concrete cover. Moreover, the stress or a unit applied load must be inversely proportional to the width o the beam (b c ) as well as the cover depth h. For a larger cover depth, i l min / h is ied, the same loads is applied to a larger member, so the stress will decrease proportionally. Summarizing the above, one can write the stresses per unit load in the ollowing orm: 1 ) lmin F ( σ II,unit 0 = h b h c (20) 14

15 2 ) lmin F ( τ II,unit 0 = h b h c (21) where b c and h are dimensionless that are the relative ratios to 1m. Through a systematic inite element analysis, the unctions F 1 and F 2 can be numerically obtained. In practical design, with the known values o lmin, b c and h, the F 1 and F 2 values can be calculated rom the ollowing statistical equations, l min l min l F , min 1 = + h h h 3 (22a) F 1 = 3.7, l min > 3 h (22b) l min l min l F , min 2 = + 3 (23a) h h h F 2 = l min > 3 h (23b) The complete solutions or the vertical normal and shear stresses in concrete near the tension rebar closest to the cut o point o FRP strips ( σ 0 andτ 0 ), can be determined by superposition: M σ II 0 ( h ) b t II, unit 0 = σ 0 = Ψ σ 0 (24) I V M τ I II 0 b 0 b, 0 = τ 0 + τ 0 = τ 0 Ib c I ( h ) t + Ψ ( h ) t II unit. (25) The ailure criterion or concrete cover separation ailure is that when the maimum principle tensile stress σ 0, 1 in concrete near the tension rebar closest to the cut o point o FRP strips is greater than the ultimate tensile strength o concrete t, ailure occurs. σ 0, 1 can be obtained by the classical stress transormation equations or a plane stress condition, 15

16 2 σ 0 σ 0 σ 0,1 = + + ( τ 0 ) 2. (26) 2 2 And t was deined in ACI code (1999) as ollows, 0.53 t = c, (27) I a strengthened RC beam is subjected to our point bending, M 0 and V 0 in terms o the totally applied load, 2P, are given M 0 = PL s ; V 0 = P. (28) Consequently, P can be determined as P = ΨL s ( h ) b II, unit t σ 0 2I + ΨL s 1 + Ib c 0.53 c ( h ) 2 b II, unit t σ 0 2I L s I ( h ) t + Ψ ( h ) b 2 b II, unit t τ 0 (29) 3. Procedure or constructing the ailure diagram In this paper, the authors attempt to draw a ailure diagram to predict the ailure mode or a given strengthened RC beam. There are ive possible ailure modes include: (a) rupture o FRP strips; (b) compression ailure ater yielding o steel; (c) compression ailure beore yielding o steel; (d) delamination o FRP strips due to crack; and (e) concrete cover separation. From the practical point o view, the thickness o FRP is a sensitive and important actor that will aect the ultimate ailure mode. With gradually increasing FRP thickness to strengthen a RC beam, the probable order or ailure occurrence is rupture o FRP, 16

17 delamination o FRP, concrete cover separation and then compression ailure. As a result, it is reasonable to set thickness o FRP (t ) as a variable, which inluences the ultimate ailure mode. Another important variable is the distance rom support to cut o point o FRP strips (L -s ), although only concrete cover separation ailure is associated with this parameter. For a particular beam to be strengthened and a given FRP material, t and L -s are the only parameters governing the ailure diagram. To identiy the ailure mode o a strengthened RC beam, the maimum strain in concrete or FRP at ailure is calculated or each individual ailure mode. The actual ailure mode is the one that gives rise to the lowest ailure strain. ( ε u When rupture o FRP strips occurs, the ailure strain is the ultimate aial strain in FRP ) obtained rom manuacturer or measurement. For compression ailure whether it occurs beore or ater steel yielding, the ailure strain is the concrete ultimate strain ( ε cu ), which is taken to be in general. Considering the delamination o FRP strips due to crack, the maimum corresponding strain in FRP ( ε d ) and strain in concrete ( ε d c ) at ailure are obtained rom Eqns. (14) and (15) as c d 2 b / bc ε = b / b (30) + c E t d 2 b / bc c ε c = 1.1 b b. (31) 1+ / c E t h For concrete cover separation ailure, in terms o Eqn. (29), we can get the maimum corresponding strain in FRP ( ε p ) and strain in concrete ( ε p c ) at ailure below, 17

18 Ω = ΨL s ( h ) b II, unit t σ 0 2I + ΨL s 1 Ibc 2 b II, unit t σ 0 + 2I b L 2 s II, unit b t τ 0 I (32) ( h ) ( h ) t + Ψ ( h ) ε p = Ω E b t d 0.53 c LL s d ( h 0.5β) + ( ) Es As d 0.5β + Es As 0.5β d h h (33) ε c p = Ω E b t d Es As h 0.53 c LL s ( h 0.5β) + ( d 0.5β) d + Es As d 0.5β h ( h ) (34) where L L-s is the distance rom the support to the loading point, and and I tr according to t can be obtained as shown in Eqns. (35) and (36). = Es E + Es E Es As + E Es As + E As + b t 2 As + b t + 2 Ecbc E Ecbc E Es E Asd + Es E d As + b t d (35) Itr 2 = Ec 3 /3 ( ) 2 Es Es bc + As d + As d + b t ( d ) 2 (36) E E E The procedure or ailure diagram construction is described in detail as ollows: 18

19 1) Firstly, the critical FRP thickness, t r-c separating FRP rupture and compression ailure ater yielding o steel and t ca-cb separating compression ailure ater yielding o steel and compression ailure beore yielding o steel, are given by: t r c = A min b (37) t ca cb = A ma b (38) r-c where A,min and A,ma are given by Eqns. (5)-(10). When FRP thickness (t ) eceeds t, the ailure mode changes rom rupture o FRP to compression ailure ater yielding o steel. ca-cb While t continues to increase to t, compression ailure beore yielding o steel may take the place o compression ailure ater yielding o steel. 2) Secondly, the occurrence o crack-induced delamination o FRP strips is analysed. Setting d dl dl r-c dl ε = ε u, we can get t using Eqn. (30). I t t, it means that when t increases to t the ailure mode changes rom rupture o FRP to delamination o FRP. I t r-c <0 or t dl > t r-c, dr t can be obtained rom Eqn. (31), with the assumption o ε d c delamination o FRP strips starts to occur in place o compression ailure. dr = ε cu. When t reaches t, 3) Lastly, concrete cover separation ailure is considered. As mentioned above, thickness o FRP (t ) and the distance rom support to cut o point o FRP strips (L -s ) are set as variables, with t as horizontal ais and L -s as vertical ais. Considering our point bending test, most cases show that L -s is not allowed to be longer than L L-s, the distance rom the support to the loading point, which means that the cut o point o FRP must be outside the constant moment region. Two situations should be considered. Fig. 6 (a) and (b) show a typical ailure diagram, or t dl t r-c and t dl > t r-c, respectively. 19

20 dl r-c When t t in the second stage, the occurrence o concrete cover separation is dl divided into two parts, namely the let part and the right part relative to t. Setting ε p = ε u, the relationship o L -s and t is obtained rom Eqn. (33), and the upper region o the curve on the let side o t dl is the let part. In order to predict the right part, the comparison betweenε d and ε p have to be done. By assuming ε d = ε p, one can get the transer curve o L -s and t rom delamination o FRP to concrete cover separation. dl Consequently, the upper region o the curve o L -s vs t on the right side o t is the right part, reerring to Eqns. (30) and (33). dl r-c When t > t in the second stage, the occurrence o concrete cover separation is r-c r-c divided into three parts, namely the let part (let o t ), the middle part (between t and dr dr t ) and the right part (right o t ). Setting ε p = ε u, the transer curve o L -s and t is obtained rom Eqn. (33), and the upper region o the curve on the let side o t r-c is the let part. In comparison, the upper region o the curve o L -s vs t between t r-c and t dr is the middle part, reerring to Eqn. (34) on account o ε c p = ε cu. Furthermore, with ε d c = ε p c, one can obtain the transer curve o L -s and t rom delamination o FRP to concrete cover separation reerring to Eqns. (31) and (34), and thus the right part is determined as the upper part o the curve o L -s vs t on the right side o t dr. 4. Derivation o the ailure diagram a speciic eample Several simply supported beams under our point bending [34] are employed as eamples to demonstrate the establishment o the ailure diagram. Appendi B presents the beam dimensions and material properties, as well as the ailure mode and ultimate load. The 20

21 establishment o ailure diagram or this particular case is shown in the ollowing, and the results are shown in Fig. 7(g). 1) Firstly, determine α 1 and β 1 rom Eqns. (3) and (4): Then, using Eqns. (5)-(7) and (37), we can get α1 = = > 0.6. β1 = = > 0.6 hε cu = ε + ε cu u = = 0.034m ε d s = = = < = ε sy ε d s = = = > = ε sy. Thereore, r c t α1 = c bc β1 + Esε s As b E ε u s A = s = 0.79 mm Net, in terms o Eqns. (8)-(10) and (38), one can obtain dε cu = ε + ε cu sy = = 0.076m and, ε = h = = ε d s = = = > = ε sy As a result, one can get ca cb t α1 = c b β + c b 1 E y ε A s = y A s. = 8.05 mm 2) Considering the occurrence o delamination o FRP strips due to crack, we can get t dl using Eqn. (30) below, 21

22 = / 0.2 t dl = 0.34 mm / 0.2 dl t Since t dl =0.34mm t r-c =0.79mm, it means that with increasing t to t dl the ailure mode changes rom rupture o FRP to delamination o FRP, without chance to ail with compression ailure. dl r-c 3) Since t =0.34mm t =0.79mm in the second stage, the occurrence o concrete cover dl separation is divided into two parts, namely the let part and the right part relative to t. Setting ε p = ε u, Eqn. (33) is changed to as ollows, = t Ω ( ) ( ) ( ) The upper region o the curve on the let side o t dl (0.34mm) is the let part. In order to predict the right part, with the assumption oε d = ε p, one can get the transer curve o L -s and t rom delamination o FRP to concrete cover separation reerring to Eqns. (30) and (33), as given below, b 1+ b / b / b c c c E t 0.53 c LL s = d E b t Es As h Ω d d Es As β h ( h 0.5β) + ( d 0.5β) and 22

23 / / t dl = t Ω ( ) + ( ) ( ) Consequently, the upper region o the curve o L -s vs t on the right side o t dl (0.34mm) is the right part. 5. Veriication and Discussions In order to veriy the applicability o the ailure diagram, published eperimental results pertaining to strengthened RC beams are analysed. Totally, 33 samples are selected rom eight reerences, showing results that cover various ailure modes. Two series o tests carried out by Nguyen et al. [35] and Fanning and Kelly [36] ocused on the eect o the L -s on the strengthening perormance. The other papers investigated the inluence o the thickness o FRP strips. The various values o L -s and t, the ultimate loads and the ailure modes rom eperiments and theoretical models in ailure diagram as well as detail inormation or all samples collected are summarised in Appendi B. The corresponding ailure diagrams or the eight groups o tests are shown in Fig. 7. The comparison between eperiments and prediction by ailure diagram shows that this method could predict the ailure mode or a strengthened RC beam. Also, the ultimate load capacity can be calculated by individual theoretical epression, ater the particular ailure mode is obtained. The ailure diagram shows that ailure mode or a strengthened RC beam may vary rom rupture o FRP strips, to delamination o FRP strips, and then to concrete cover separation, with increasing FRP thickness. With the correct design, compression ailure 23

24 ater yielding o steel may take place beore local ailure. Reducing the distance rom support to cut o o FRP may decrease the likelihood o concrete cover separation. From the practical application point o view, compression ailure ater yielding o steel is most preerable in design. However, the occurrence o local ailures such as delamination o FRP strips and concrete cover separation precludes the chance o compression ailure ater yielding o steel. In order to have compression ailure ater steel yielding, besides reducing the distance rom support to cut o o FRP as ar as possible, appropriate selection o FRP properties is very important. FRP with good perormance, such as high strength, high elongation at ailure and high modulus, may not be eective in practical applications, because ailure may occur by delamination o FRP strips early with low aial strain in FRP. Beore closing, a ew remarks should be made on the use o the ailure diagram in practice. To perorm strengthening o a given RC beam, the beam dimensions and reinorcement ratio are ied. Also, the selection o FRP properties is perhaps limited by the availability o commercial products. Consequently, the variables to be chosen are only the FRP dimensions, including the FRP thickness, length and width. For a particular concrete beam, the FRP width can be selected as a certain percentage o the beam width. The two parameters represented on the ailure diagram are then suicient to determine the ailure mode. Ater knowing the possible ailure mode, the load capacity and delection o the beam can be accurately predicted. Since the ailure diagram summarizes all possible ailure modes, a clear picture o all possibilities are provided to guide the designer in choosing the best combination o plate thickness and length. The plotting o ailure diagrams will also acilitate the selection o the best material. Moreover, with the ailure mode predicted, the critical ailure initiation location can be known. That inormation is very useul or the continuous monitoring o strengthened beams, as well as the determination o appropriate positions or the application o anchors. 24

25 The idea behind the ailure diagram is that the ailure mode associated with the lowest strain in FRP or concrete by comparison is most likely to occur. In this paper, a general concept is proposed. With uture development leading to better methods or local ailure, the equations in this paper can be urther reined. 6. Conclusions Numerous studies including eperimental research, theoretical analysis and numerical simulation have demonstrated that epoy bonding o ibre reinorced plastic (FRP) strips to the tension soit o reinorced concrete (RC) beams can signiicantly improve the ultimate leural strength and stiness. Several important ailure modes have been studied, such as compression ailure beore or ater yielding o steel, rupture o FRP strips, delamination o FRP strips and concrete cover separation. This paper attempts to build a ailure diagram to show the relationship and the transition among dierent ailure modes or RC beams strengthened with FRP strips, and how ailure modes vary with FRP thickness and the distance rom the end o FRP strips to the support. The idea behind this ailure diagram is that the ailure mode associated with the lowest strain in FRP or concrete by comparison is most likely to occur. By comparison between predictions based on ailure diagram and eperimental results, we show that this method could predict the ailure mode or a strengthened RC beam. Knowing the ailure mode, the ultimate load capacity can be calculated. The ailure diagram provides guidelines to practical design, and is useul in establishing a procedure or selecting the type and size o FRP or the eternal strengthening o RC beam. Acknowledgements 25

26 The Research Grants Council o the Hong Kong SAR (Project No. HKUST 6050/99E), provided the inancial support o this work. The authors wish to thank the Construction Materials Laboratory, Advanced Engineering Material Facilities, and Design and Manuacturing Services Facility in HKUST or their technical supports. Reerences [1] Buyukozturk O, Hearing B. Failure behaviour o precracked concrete beams retroitted with FRP. J. Compos. Constr. 1998; 2(3): [2] EI-Mihilmy MT, Tedesco JW. Analysis o reinorced concrete beams strengthened with FRP laminates. J. Struct. Eng. 2000; 126(6): [3] EI-Mihilmy MT, Tedesco JW. Prediction o anchorage ailure or reinorced concrete beams strengthened with iber-reinorced polymer plates. ACI Struct. J. 2001; 98(3): [4] Chaallal O, Nollet MJ, Perraton D. Shear strengthening o RC beams by eternally bonded side CFRP strips. J. Compos. Constr. 1998; 2(2): [5] Triantaillou TC. Shear strengthening o reinorced concrete beams using epoy-bonded FRP composites. ACI Struct. J. 1998; 95(2): [6] Triantaillou TC, Antonopoulos CP. Design o concrete leural members strengthened in shear with FRP. J. Compos. Constr. 2000; 4(4): [7] Norris T, Saadatmanesh H, Ehsani MR. Shear and leural strengthening o R/C beams with carbon iber sheets. J. Struct. Eng. 1997; 123(7): [8] Mitsui Y, Murakami K, Takeda K, Sakai H. A study on shear reinorcement o reinorced concrete beams eternally bonded with carbon iber sheets. Compos. Interace 1998; 5(4): [9] Pellegrino C, Modena C. Fiber reinorced polymer shear strengthening o reinorced concrete beams with transverse steel reinorcement. J. Compos. Constr. 2002; 6: [10] Triantaillou TC, Plevris N. Strengthening o RC beams with epoy bonded ibre composite materials. Mater. Struct. 1992; 25: [11] Rabinovich O, Frostig Y. Closed-orm high order analysis o RC beams strengthened with FRP strips. J. Compos. Constr. 2000; 4(2):

27 [12] Shen HS, Teng JG, Yang J. Interacial stresses in beams and slabs bonded with thin plate. J. Eng. Mech. 2001; 127(4): [13] Oehlers DJ, Moran JP. Premature ailure o eternally plated reinorced concrete beams. J. Struct. Eng. 1992; 116(4): [14] Ahmed O, Gemert DV. Eect o longitudinal carbon iber reinorced plastic laminates on shear capacity o reinorced concrete beams. In: Proc. 4th Int. Symp. Fiber Reinorced Polymer Reinorcement or Reinorced Concrete Structures. Maryland, USA, p [15] Roberts TM. Approimate analysis o shear and normal stress concentrations in the adhesive layer o plated RC beams. Structural Eng. 1989; 67(12): [16] Varastehpour H, Hamelin P. Strengthening o concrete beams using iber-reinorced plastics. Mater. Struct. 1997; 30: [17] Malek AM, Saadatmanesh H, Ehsani MR. Prediction o ailure load o R/C beams strengthened with FRP plate due to stress concentration at the plate end. ACI Struct. J. 1998; 95(1): [18] An W, Saadatmanesh H, Ehsani MR. RC beams strengthened with FRP plates II: analysis and parametric study. J. Struct. Eng. 1991; 117(11): [19] Chaallal O, Nollet MJ, Perraton D. Strengthening o reinorced concrete beams with eternally bonded iber-reinorced-plastic plates: design guidelines or shear and leure. Canadian J. Civil Eng. 1998; 25: [20] Saadatmanesh H, Malek AM. Design Guidelines or Strengthening o RC Beam with FRP Plates. J. Comp. Constr. 1998; 2: [21] Ross CA, Jerome DM, Tedesco JW, Hughes ML. Strengthening o reinorced concrete beams with eternally bonded composite laminates. ACI Struct. J. 1999; 96: [22] Almusallam TH, Al-Salloum YA. Ultimate strength prediction or RC beams eternally strengthened by composite materials. Compos. Part B 2001; 32: [23] Ng SC, Lee S. Analysis o leural behavior o reinorced concrete beam strengthened with CFRP, Proc. 13 th Int. Con. Compos. Mater., Beijing, China, ID [24] Leung CKY. Delamination ailure in concrete beams retroitted with a bonded plate. J. Mater. Civil Eng. 2001; 13:

28 [25] Teng JG, Chen JF, Smith ST, Lam L. FRP-Strengthened RC Structures. John Wiley & Sons, LTD, [26] Neubauer U, Rostasy FS. Design aspects o concrete structures strengthened with eternally bonded CFRP plates. Proc. Seventh Int. Con. on Structural Faults and Repairs, Edinburgh, UK, p [27] Yuan H, Wu Z. Interacial racture theory in structures strengthened with composite o continuous iber. Proc. Symp. China and Japan, Sci. & Technol. o 21 st Century, Tokyo, Japan, p [28] Ziraba YN, Baluch MH, Basunbul IA, Shari AM, Azad AK, Al-Sulaimani GJ. Guideline toward the design o reinorced concrete beams with eternal plates. ACI Struct. J. 1995; 91: [29] Roberts TM, Haji-Kazemi H. Theoretical study o the behaviour o reinorced concrete beams strengthened by eternally bonded steel plates. Proc. Instn Civ. Engrs., Part ; 87: [30] Lau KT, Dutta PK, Zhou LM, Hui D. Mechanics o bonds in an FRP bonded concrete beam. Compos. Part B 2001; 32: [31] Raoo M, Zhang S. An insight into the structural behaviour o reinorced concrete beams with eternally bonded plates. Proc. Instn Civ. Engrs Structs & Bldgs 1997; 122: [32] Zhang S, Raoo M, Wood LA. Prediction o peeling ailure o reinorced concrete beams with eternally bonded plates. Proc. Instn Civ. Engrs Structs & Bldgs 1997; 122: [33] Gao B, Leung CKY, Kim JK. Prediction o concrete cover separation ailure or RC beams strengthened with CFRP strips. Eng. Struct. 2005; 27; [34] Rahimi H, Hutchinson A. Concrete beams strengthened with eternally bonded FRP plates. J. Compos. Constr. 2001; 5: [35] Nguyen DM, Chan TK, Cheong HK. Brittle ailure and bond development length o CFRP concrete beams. J. Compos. Constr., 2001; 5: [36] Fanning PJ, Kelly, O. Ultimate response o RC beams strengthened with CFRP plates. J. Compos. Constr. 2001; 5:

29 [37] Alagusundaramoorthy P, Harik IE, Choo CC. Fleural behavior o R/C beams strengthened with carbon iber reinorced polymer sheets o abric. J. Compos. Constr. 2003; 7: [38] Arduini M, Tommaso AD, Nanni A. Brittle ailure in FRP plate and sheet bonded beams. ACI Struct. J. 1997; 94(4): [39] Gao B, Kim JK, Leung CKY. Fracture behavior o RC beams with FRP strips bonded with rubber modiied resins: eperiment and FEM model. Compos. Sci. Technol. 2004; 64: [40] Maalej M, Bian Y. Interacial shear stress concentration in FRP-strengthened beams. Compos. Struct. 2001; 54:

30 Appendi A. Prediction o the Ultimate Fleural Strength o Strengthened RC Beams Generally speaking, two situations should be considered, including a) obtaining the quantity o FRP to satisy the requirement o moment capacity or a RC beam, and b) calculating the moment capacity o a strengthened RC beam. Both cases are the reverse processes. From the practical design point o view, the ormer is the more common case. To archive the targeted moment, M u, the ollowing ormula is employed, Mu = α ( ) ( ) + 1 c b c β1 h 0.5β1 y A s h d y A s h d (39) Only one unknown variable,, eists in the equation, which can be obtained as the solution o a quadratic equation, 2 1 a1 b1 b 4a1c1 = (40) 2 a 2 1 = 0.5α1β1 c b c (41) b 1 = α1β1 c b c h (42) c = + ( ) 1 Mu y As h d y As h d. (43) Knowing, we can get, ε s, ε s, andε by linear strain distribution withε c =0.0035, ε d s = (44) ε d s = (45) d ε = (46) I ε s > ε sy and ε < ε u, the FRP area is within the range A,min A A,ma. In this case, compressive ailure will occur ater steel yielding. A can be obtained rom the Eqns. (9) or (10), as well as M u rom Eqn. (39). Iε s < εsy and ε < ε u, the yielding o steel in tension 30

31 can not be obtained beore ailure. Thereore, in order to have enough ductility and warning beore ailure, the area o FRP must be reduced to A,ma. The ultimate moment resistance, M u is then lower than M u. The calculation o A,ma has been introduced in Eqns. (8), (9) and (10). Then, we can get M u, as ollows, M ( ) ( ) d u = α c bc β h β y As h d + Esε s As h d ε , s = < ε sy (47) M ( ) ( ) d u = α c b c β h β y A s h d + y A s h d ε , s = ε sy (48) When ε > ε u, rupture o FRP strips is to occur instead o compression ailure. In this case, the area o FRP can be increased to a value within the range o A,min and A,ma, to assure the occurrence o compression ailure ater yielding o steel. The ultimate moment resistance M u, which can be obtained using Eqns. (51) and (52), will then be higher than M u. To calculate the moment capacity o a given strengthened RC beam, two balanced limited values o cross section area o FRP, A,min and A,ma are calculated irst, to determine the ailure mode in terms o A. For each individual ailure mode, can be obtained rom orce equilibrium o the cross section and the relationship among strain components (Eqns. (44)- (46)). The ultimate moment resistance M u, can be calculated by the equations below or dierent situations. When A < A,min, M u M u = A E d ε u < h (49) = A E ε u When A,min A A,ma, M u ( h 0.5β ) y As ( d β ) Esε s As β d , ε s = ε u ε sy ( h 0.5β ) + y A s ( d 0.5β ) + y A s 0.5β d, ε s = ε u ε sy d h (50) = α d c bc β < ( h 0.5β ) y As ( h d ) + Esε s As h d, ε s = ε sy (51) 31

32 32 ( ) ( ) sy d s d h A s y d h y A s h b c c u M ε ε β β α = + = , (52) When A >A,ma, ( ) ( ) sy d s d h A s s E s d h s A s E s h b c c u M ε ε ε ε β β α < = + = , (53) ( ) ( ) sy d s d h A s y d h s A s E s h b c c u M ε ε ε β β α = + = , (54)

33 Appendi B. Details o Eperiments Eperiments Alagusundaramo orthy et al. [37] Arduini et al. [38] Fanning and Kelly [36] Beam Beam Beam FRP Beam width depth length length (mm) (mm) (mm) (mm) FRP FRP A s A s d width thickness (mm) (mm) (mm) d L -s (mm) (mm) CB11-1F Φ 25 2 Φ CB11-1F Φ 25 2 Φ CB11-2F Φ 25 2 Φ CB11-2F Φ 25 2 Φ A Φ 14 2 Φ A Φ 14 2 Φ A Φ 14 2 Φ FKF Φ 12 2 Φ FKF Φ 12 2 Φ FKF Φ 12 2 Φ FKF Φ 12 2 Φ T Φ 10 2 Φ Gao et al. [39] T Φ 10 2 Φ T Φ 10 2 Φ T Φ 10 2 Φ MB Φ 10 2 Φ Maalej and Bian [40] MB Φ 10 2 Φ MB Φ 10 2 Φ MB Φ 10 2 Φ A Φ 10 2 Φ Nguyen et al. [35] A Φ 10 2 Φ A Φ 10 2 Φ A Φ 10 2 Φ Rahimi and Hutchinson [34] Triantaillou and Plevris [10] RHB Φ 10 2 Φ RHB Φ 10 2 Φ RHB Φ 10 2 Φ RHB Φ 10 2 Φ Φ Φ Φ Φ Φ Φ

34 Appendi B. (Continued) Eperiments Beam c t E c E s E P model P ep (MPa) (MPa) (GPa) (GPa) (GPa) (kn) (kn) a Failure a Failure mode model mode ep CB11-1F RF RF Alagusundaramoorthy CB11-1F RF RF et al. CB11-2F DF DF [37] CB11-2F DF DF A CS CS Arduini et al. A CS CS [38] A CS CS FKF CS CS Fanning and Kelly FKF CS CS [36] FKF CS CS FKF CS CS T RF RF Gao et al. [39] T DF DF T DF CS T CS CS MB CC RF Maalej and Bian [40] MB CC CS MB CS CS MB CS CS A CS CS Nguyen et al. A CS CS [35] A CS CS A DF CC RHB DF DF Rahimi and RHB DF DF Hutchinson RHB CS CS [34] RHB CS CS RF RF DF DF Triantaillou DF DF and Plevris DF DF [10] DF DF CS DF a CC = Compression ailure; RF = Rupture o FRP strips; DF = Delamination o FRP strips; CS = Concrete cover separation 34

35 Figure Captions Fig. 1: Failure modes o FRP strengthened RC beams: (a) Compression ailure; (b) Rupture o FRP strips; (c) Shear ailure; (d) Delamination o FRP strips; and (e) Concrete cover separation Fig. 2: Cross section dimensions with strain distribution and stress diagram Fig. 3: Analysis in stage II with opposite aial orce in FRP strips Fig. 4: The comparison o prediction with/without modiication actor Fig. 5: FEM models or predicting σ II,unit 0 FRP; and (c) 2-D without FRP andτ II,unit 0 : (a) 3-D with FRP; (b) 2-D with Fig. 6: A typical ailure diagram in the third step: (a) t dl t r-c and (b) t dl > t r-c (using t dr ) Fig. 7: Demonstration o established ailure diagram compared to eperiments done by: (a) Alagusundaramoorthy et al., 2003; (b) Arduini et al. 1997; (c) Fanning and Kelly, 2001; (d) Gao et al., 2003; (e) Maalej and Bian, 2001; () Nguyen et al., 2001; (g) Rahimi and Hutchinson, 2001; and (h) Triantaillou and Plevris,

36 (a) (b) (c) (d) (e) Fig. 1. Failure modes o FRP strengthened RC beams: (a) Compression ailure; (b) Rupture o FRP strips; (c) Shear ailure; (d) Delamination o FRP strips; and (e) Concrete cover separation 36