Shear Strength Model for FRP-Strengthened RC Beams with Adverse FRP-Steel Interaction G. M. Chen 1 ; J. G. Teng 2 ; and J. F.

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1 Ti i te Pre-Publied Verion. Sear Strengt Model or FRP-Strengtened RC Beam wit Advere FRP-Steel Interaction G. M. Cen 1 ; J. G. Teng 2 ; and J. F. Cen 3 Abtract: RC beam ear-trengtened wit externally-bonded FRP U-trip or ide trip uually ail due to debonding o te bonded FRP ear reinorcement. Becaue uc debonding uually occur in a brittle manner at relatively mall ear crack widt, ome o te internal teel tirrup interected by te critical ear crack may not ave reaced yielding at beam ear ailure. Conequently, te yield trengt o internal teel tirrup in uc a trengtened RC beam cannot be ully utilized. Ti advere ear interaction between te internal teel ear reinorcement and te external FRP ear reinorcement may igniicantly reduce te beneit o te ear-trengtening FRP but a not been conidered explicitly by any o te ear trengt model in te exiting deign guideline. Ti paper preent a new ear trengt model conidering ti advere ear interaction troug te introduction o a ear interaction actor. A compreenive evaluation o te propoed model, a well a tree oter ear trengt model, i conducted uing a large tet databae. It i own tat te propoed ear trengt model perorm te bet among te model compared, and te perormance o te oter ear trengt model can be igniicantly improved by including te propoed ear interaction actor. Finally, a deign recoendation i preented. CE Databae ubject eading: Fiber reinorced polymer; Reinorced Concrete; Concrete beam; Bonding; Sear ailure; Sear reitance; Sear trengt. 1 Potdoctoral Fellow, Department o Civil and Structural Engineering, Te Hong Kong Polytecnic Univerity, Hong Kong, Cina 2 Cair Proeor o Structural Engineering, Department o Civil and Structural Engineering, Te Hong Kong Polytecnic Univerity, Hong Kong, Cina (Correponding autor); cejgteng@polyu.edu.k 3 Reader, Intitute or Inratructure and Environment, Scool o Engineering, Te Univerity o Edinburg, Edinburg, U.K. 1

2 INTRODUCTION Te external bonding o iber-reinorced polymer (FRP) to reinorced concrete (RC) tructure a become a popular trengtening tecnique in te pat decade; te tecnique a alo received muc reearc attention (Bank 26; Hollaway and Teng 28; Oeler and Seracino 24; Teng et al. 22). In particular, te ear reitance o RC beam can be enanced by bonding FRP ear reinorcement in te orm o complete wrap, U-jacket and ide trip (Cen and Teng 23a, b; Teng et al. 22). Witout lo o generality, te FRP ear reinorcement i aumed erein to be in te orm o dicrete trip or eae o dicuion; a continuou eet wit ibre oriented in a ingle direction can be treated a dicrete trip in te ibre direction wit a zero net gap between trip. Exiting reearc a etablied a general picture o te tructural beaviour o RC beam ear-trengtened wit FRP and led to a number o ear trengt model or tem (Cen and Teng 23a, b; Kalia et al. 1998; Monti and Liotta 27; Triantaillou 1998; Triantaillou and Antonopoulo 2); te more reliable o tee ear trengt model ave been adopted by deign guideline (ACI-44.2R 28; CNR-DT2 24; ib 21; HB35 28). A compreenive review o exiting work (Cen 21), owever, reveal tat everal apect o te beaviour o uc trengtened beam are till not well undertood. In particular, te advere interaction between te dierent component o ear reitance (Ali et al. 26; Cen et al. 21; Pellegrino and Modena 22, 26, 28) a been identiied a a major iue tat require urter reearc. Ti paper deal wit te eect o interaction between te internal teel ear reinorcement (only tirrup are conidered to impliy te problem) and te external FRP ear reinorcement in RC beam ear-trengtened wit FRP U-trip or ide trip. Suc trengtened beam coonly ail due to te debonding o FRP trip rom te beam ide (Cen and Teng 23b; Teng and Cen 29). Ti ailure mode i uually brittle o tat te widt o te critical ear crack i limited wen FRP debonding occur (Ali et al. 26; Cen et al. 21; Pellegrino and Modena 28). A a reult, at te intance o debonding ailure, te component o ear reitance rom concrete i likely to be well maintained (e.g. Bouelam and Caallal 28), but te component o ear reitance rom teel may be igniicantly below wat i expected in a conventional RC beam becaue not all teel tirrup in an FRP-trengtened RC beam interected by te critical ear crack can reac yielding at te ear ailure o te beam (Ali et al. 26; Cen et al. 21; Deniaud and Ceng 21; Li et al. 22; Monti and Liotta 27; Pellegrino and Modena 28; Teng et al. 22; Teng et al. 24). It may be noted tat ti advere ear interaction eect a not been duly conidered in any o te exiting deign guideline (Cen 21). Sear trengt model in exiting guideline are baed on te imple additive approac tat te ear reitance o a ear-trengtened RC beam can be ound rom te ollowing equation:: Vu = Vc + V + V (1) were V c, V and V are te component contributed by te concrete, te teel ear reinorcement, and te FRP ear reinorcement repectively. Te value o V c and V are generally evaluated uing proviion in exiting deign code or RC tructure, wile variou expreion ave been propoed or V. Eq. (1) implie tat te tree ear reitance component reac teir ultimate value imultaneouly in a real beam, wic i over-optimitic and un-conervative. A number o tudie ave been conducted to conider te ear interaction iue (e.g. Ali et al. 26; Li et al. 21; Pellegrino and Modena 22, 26, 28), leading to everal ear trengt model tat conider te ear interaction eect (Li et al. 21; 2

3 Pellegrino and Modena 22, 26, 28). Tee model, owever, ave been developed on te bai o limited experimental reult and tu uer rom inevitable limitation. Recently, Modii and Caallal (211) propoed a new ear trengt model tat account or ti advere ear interaction eect by introducing te o-called cracking modiication actor (β c ) wic i related to te rigiditie o bot teel ear reinorcement and FRP ear reinorcement; te expreion o β c wa determined by curve-itting o te experimental reult or V. Tee autor ave own tat te incluion o β c can improve te perormance o te propoed ear trengt model a well a ome oter ear trengt model. Wilt te work repreent a valuable tep orward in undertanding and modeling te FRP-teel interaction eect, teir model require improvement, epecially or beam wit FRP U-trip were te FRP ear contribution (V ) i igniicantly overetimated or a large number o pecimen. To undertand te interaction between te tree component o ear reitance in Eq. (1), it i neceary to invetigate ow eac o tem develop during te loading proce. I tee component are quantiied during te loading proce, te ear reitance o te beam can alo be quantiied trougout te loading proce and it ultimate value can be obtained by inding te maximum o te um o te tree component a cematically own in Fig. 1. Te autor ave recently employed a teoretical approac to etabli te development o ear contribution rom te FRP (Cen et al. 211) and te teel tirrup (Cen et al. 21) trougout te loading proce a caracterized by te critical ear crack widt. Ti paper irt preent a ear trengt model or FRP debonding ailure conidering te advere FRP-teel ear interaction developed baed on te work preented in Cen et al. (21; 211). It perormance i ten aeed uing a large tet databae collected rom te literature. A impliied deign recoendation i inally preented. SHEAR STRENGTH MODEL ACCOUNTING FOR FRP-STEEL INTERACTION A wit mot o te ear trengt model in exiting guideline, te propoed ear trengt model i baed on te aumption tat te ear ailure o an FRP ear-trengtened RC beam i dominated by a ingle critical ear crack a cematically own in Fig. 2, and te ear contribution o bot FRP trip and teel tirrup can be evaluated by tru analogy. A dicued earlier, te contribution rom te concrete, internal teel tirrup and external FRP trip develop gradually during te loading proce (Fig. 1). For FRP debonding ailure, it may be aumed tat te contribution o concrete to te ear capacity o te beam (V c ) i te ame a tat in an un-trengtened RC beam becaue te widt o te critical ear crack i likely to be mall wen te beam ail due to FRP debonding (Bouelam and Caallal 28). Tereore, te ear reitance o te beam can be expreed a [intead o Eq. (1)]: Vu = Vc + KV, p + K V, p (2) were V, p and V, p are te maximum ear contribution o teel tirrup and FRP trip repectively, K and K are mobilization actor or te teel tirrup and FRP trip repectively wic ave been deined by Cen et al. (21) a: K = σ e, / y (3) K = σ, e/, e (4) in wic σ,e and σ,e are repectively te average tre in te teel tirrup and FRP trip interected by te critical ear crack, y i te yield trengt o te teel tirrup, and,e i te eective (average) tre in te FRP interected by te critical ear crack wen V peak (i.e. V = V, p, wic doe not necearily correpond to te ultimate tate o te 3

4 beam a own in Fig. 1). K and K are repectively proportional to te average tre in te teel tirrup and tat in te FRP trip, wic are in turn directly related to te ear crack widt w. Clearly, K and K relect te degree o mobilization o te teel tirrup and tat o te FRP trip repectively in reiting ear at a given load level or a ear crack widt and capture te interaction between teel tirrup and FRP trip in reiting ear. Development o te FRP Contribution K V wit crack widt It all be noted tat in bot te numerical tudy (Cen et al. 21) and analytical olution (Cen et al. 211) on wic te preent tudy i baed, it wa aumed tat te widt o te critical ear crack varie linearly rom te crack tip to te crack end; ti aumption normally lead to conervative reult or bot FRP trip and teel tirrup (Cen 21). Wit ti aumption, te maximum value o te ear crack widt i alway at te crack end (Fig. 2); ti value i reerred to a te crack end widt and i repreented by w e in ti paper. In Cen et al. (21, 211), it wa alo aumed tat te upper end (i.e. te crack tip) o te critical ear crack at te ultimate tate i located at.1d rom te compreion ace o te beam (ee Fig. 2), wit d being te eective dept o te beam. Baed on te above aumption and te ull-range beaviour o FRP-to-concrete bonded joint, Cen et al. (211) developed cloed-ormed olution or te development o te ear contribution o FRP ( V ) wit te crack end widt ( w e ) or bot FRP U-trip and ide trip. Figure 3 ow example V we curve, were te peak load are denoted by P u and P repectively or FRP U-trip and ide trip. From te V we curve, te K we curve can be eaily obtained according to Eq. (3). Figure 4 ow te K we curve correponding to te V we curve in Fig. 3. Cen et al. (211) demontrated tat te K we curve depend mainly on te FRP tine Et and te beam eigt (wic can be repreented by te eective eigt o FRP, e a own in Fig. 2). Te maximum FRP contribution V, p (te peak value on te V we curve) can be obtained by etting V we = baed on te V we relationip preented in Cen et al. (211). Te general expreion or V, p i given a (Cen et al. 211):, e( cotθ + cot β) in β V, p= 2, etw (5) = σ D (6), e,max rp σ, max = min (7a) σ db,max σ db,max = 2E G t π in 2 L L e 2E t G L L max max L e < L e (7b) 4

5 , e + t + b or ide trip 2in β L = (7c) max + +, e t b or U - trip in β were, e i te eective (average) tre in te FRP interected by te critical ear crack; w i te widt o individual FRP trip perpendicular to te iber direction (all FRP trip are aumed to ave te ame w ); i te centre-to-centre pacing o FRP trip meaured along te longitudinal axi o te beam (te FRP trip are aumed to be evenly ditributed; and tu or an FRP continuou eet, = w in β.); t i te FRP trip tickne; θ i te angle between te critical ear crack and te longitudinal beam axi; β i te angle between te iber direction and te longitudinal beam axi; σ,max db,max i te maximum tre in te FRP trip interected by te critical ear crack; σ i te maximum tre in te FRP trip interected by te critical ear crack a governed by debonding ailure; L max i te maximum bond lengt o FRP trip interected by te critical ear crack; L e i te eective bond lengt o FRP trip a deined by Eq. (16); i te tickne o concrete cover (rom te beam bottom to te b crack end) (ee Fig 2); t i te vertical ditance rom te top o FRP trip to te crack tip (ee Fig 2); and D rp i te tre/train ditribution actor determined according to Cen et al. (211) a ollow. For FRP ide trip, te expreion o D rp i given by π d db Drp = 1 (1 ) (8) 4, e, e, e db d = (9) π 1+ ( k 1) 2 = L in β (1) db m b L = k L (11) m e 2 π π π π (, e+ b) 2 k = π Le inβ π For FRP U-trip, te expreion o D i given by D w L d rp, e rp π d = 1 (1 ) (13) 4 = 2δ w = δ ep, ep,, e in( θ + β ) π + 2 Le inβ in( θ + β), e 1 `1 τ e = (16) δ Et t δ = 2G τ (17) (12) (14) (15) 5

6 G =.38β (18) w t τ = 1.5β (19) w t β = w t ( β ) ( β ) 2 w in 1+ w in =.395 (21).55 cu In Eq. (8)-(21), L m i te maximum mobilized bond lengt in te ibre direction rom te critical ear crack to te otening ront (ee Fig 5); db i te vertical ditance rom te crack end to te interection between te rigt mot debonded FRP trip and te critical ear crack (i.e. point N in Fig. 5); i te vertical ditance rom te crack tip to te point o te interection between te debonding ront and te critical ear crack (i.e. point M in Fig. 5); w ep, (denoted by wep, and wep, u in Fig. 3 or FRP ide trip and FRP U-trip repectively ) i te crack end widt at wic te FRP ear contribution ( V ) reace it peak value V, p (denoted by V, p and V, p u in Fig. 3 or FRP ide trip and FRP U-trip repectively); δ i te interacial lip at te ear crack at wic te FRP i ully debonded; β i te trip widt coeicient; w G and τ are te interacial acture energy and maximum interacial ear tre repectively, wic can be determined according to Lu et al. (25) bond-lip model a own in Eq. (18)-(21); i te cube compreive trengt o concrete and can be etimated cu ' rom te cylinder compreive trengt o concrete uing cu = c.8a may be deduced rom EN-BS (24). It ould be noted tat in deriving te expreion o D rp [i.e. Eq. (8) and (13)], te ollowing condition repreenting a reaonable limitation on te practical FRP coniguration and/or beam ize ould be atiied a explained in Cen et al. (211): ( t+, e)coecβ > bcoecβ and ( t+, e)coecβ > Le. K -w e Curve To tudy te ear interaction between externally bonded FRP trip and internal teel tirrup, an FE model wa propoed by Cen et al. (21) in wic appropriate bond-lip relationip were employed to accurately depict te bond beaviour o bot FRP trip and teel tirrup. Numerical reult rom Cen et al. (21) owed tat te mobilization actor K o teel tirrup depend mainly on te beam eigt a well a te diameter d φ and yield trengt y o teel bar. Te concrete trengt ( (2) ' c ) alo a ' ome eect on te K we curve, but it i rater mall witin te practical range o c value. Baed on tee obervation, te ollowing expreion or te K we curve wa developed by curve-itting baed on te numerical reult or a concrete trengt ' c = 3 MPa (Cen 21): 1.4 we K = (22) 1.4 A+ we were A i a contant relecting te eect o beam ize, teel bar diameter and yield trengt. For plain bar tirrup, 6

7 (, ) ( )( φ + ) 1.48 Ln e 4.52 y A = 1 (23) For deormed bar tirrup, 4.94 Ln(, e) 3.34 ( y 245)( φ.767) A = 1 (24) Example comparion between te prediction o te Eq. (22)-(24) and te original numerical reult are own in Fig. 6 and 7 repectively or plain and deormed bar. Clearly tee expreion are in cloe agreement wit te numerical K we curve. It ould be noted tat altoug only a limited number o cae are conidered in Fig. 6 and 7, ti concluion a been validated againt a muc larger number o K we curve covering practical range o beam eigt, bar diameter, and bar yield trengt (Cen 21). Sear Interaction Factor (K) Te equation or te ear capacity o an RC beam ear trengtened wit FRP [i.e. Eq. (2)] may alternatively be expreed a Vu = Vc + V, p + KV, p (25) were K i termed te ear interaction actor wic relect te reduction o te eiciency o te FRP trengtening due to te advere interaction eect between teel tirrup and FRP trip. Te term KV, p in Eq. (25) tu repreent te net additional ear reitance contributed by te externally bonded FRP ear reinorcement. Comparing Eq. (2) wit Eq. (25) give: Vp, K = K + ( K 1) = K + ( K 1) μ (26) V, p were μ i te ratio o te ear contribution o teel tirrup to tat o FRP trip i te eect o ear interaction i not conidered: V, p yav μ = = (27) V, p, earp in wic A v and A rp are repectively te cro-ectional area o te teel tirrup and tat o te FRP trip interected by te critical ear crack. Eq. (26) ow tat te eiciency o FRP ear trengtening i aected not only by te mobilization actor K and K, but alo by te cro-ectional area o teel tirrup relative to tat o FRP trip a implied by te deinition o μ. For a given trengtening deign, μ i known, o te development o K wit te crack end widt can be ound by ubtituting te expreion o K and K into Eq. (26). Example o te K we curve o obtained are own in Fig. 8(a) and 8(b) repectively or FRP ide trip and FRP U-trip. Clearly, an increae in μ reult in a decreae in te peak value o K, indicating tat te FRP trengtening i le eicient or beam wit eavier teel ear reinorcement. Ti trend i in agreement wit tet obervation (Bouelam and Caallal 24; Li et al. 21; Pellegrino and Modena 22, 26). At te ultimate tate, te contribution o teel tirrup and FRP trip combined reace te maximum value. Ti i equivalent to K reacing it maximum value K max 7

8 on te K we curve a own in Fig. 8. Te K max value can ten be ued in Eq. (25) or te ultimate limit tate deign. For FRP U-trip, te K max value uually occur at a crack end widt w e wen V peak (i.e. we = we, p), tu, K = K, p = 1 [ee Fig. 8(b)]. Conequently, te K max value can be obtained by ubtituting K = 1 into Eq. (26) and te correponding value o w e (i.e. we we, p = ) rom Eq. (15) (ee Fig. 3) into Eq. (22) repectively, a ollow: ( K ) Kmax = 1 μ 1 (28) in wic 1.4 wep, K = 1.4 A+ w (29) ep, were A can be obtained rom Eq. (23) and (24) repectively or plain bar and deormed bar, and wep, rom Eq. (15). For FRP ide trip, te crack end widt at wic Kmax occur, w eu,, varie wit μ [ee Fig. 8(a)]: te larger te μ, te larger te w eu,. Finding te Kmax value uing te K w e curve own in Fig. 8(a) i omewat involved becaue it in turn depend on te correponding K we and K we curve. To addre ti problem, te ollowing approximate expreion o K max a been developed baed on a regreion analyi o a large number o K max value obtained rom te above procedure covering te geometrical and material propertie o te FRP trip, teel tirrup and beam over teir repective practical range (ee te peciication below): B Kmax = (3) B + μ in wic B = ( λ ) or plain bar tirrup (31) φ y and B = ( λ ) φ or deormed bar tirrup (32) y were λ =, e Le i te normalized eective eigt o FRP on te beam ide. It ould be noted tat Eq. (3)-(32) were obtained uing te lowing range o parameter: or plain bar, y = MPa, φ = 6 1 ; or deormed bar, = MPa, φ = 8 16 ; and λ = 1 2. y Repreentative comparion between prediction o Eq. (3)-(32) and K max value obtained directly rom K we curve are own in Fig. 9(a) and 9(b) or plain bar tirrup and deormed bar tirrup repectively. Clearly tee expreion provide a cloe approximation to te K max value or FRP ide trip and are adopted in te ollowing analye. COMPARISON WITH TEST DATA To validate te propoed ear trengt model, an extenive literature review a been carried out to collect tet data o RC beam ear-trengtened wit bonded FRP. Table 1 8

9 preent te collected databae or ear-trengtened RC beam tat ailed due to FRP debonding. It contain 131 pecimen, including 78 beam ear-trengtened wit FRP ide-trip and 53 beam ear-trengtened wit FRP U-trip. Only te geometric and material propertie required to determine te contribution o FRP trip to te ear capacity o te trengtened beam by te trengt model preented in ti paper are own. Furter detail can be ound in te original ource or in Cen (21). Tet reult tat were not uiciently well documented, and toe pecimen damaged beore trengtening [e.g. pecimen F1 o Mitui et al. (1998)], wit a very low concrete ' trengt ( c <15 MPa) [ome pecimen o Monti and Liotta (27)] or aving a marginal FRP ear enancement [e.g. pecimen V4 o Sundarraja et al. (28)], ave been excluded rom te databae. Te tet data lited in Table 1 ave te ollowing parametric range: beam eigt = 11 6 ; web tickne b w = 7 3 ; cylinder compreive trengt o concrete ' c = MPa ; and teel ear reinorcement ratio ρ =..75%. Mot o tee pecimen ave a ear pan-to-dept ratio / d 2.2 ; a ew tet pecimen wit / d < 2.2 [e.g. ome pecimen o Mitui et al. (1998) wit / d < 2.2] are alo included in te tet databae becaue te FRP debonding ailure mode wa clearly oberved in tee pecimen. It all be noted tat in Table 1, i te trengtened pecimen a a dierent concrete trengt rom tat o te control pecimen, te tet ear contribution o te FRP a been adjuted uing te metod decribed by Cen and Teng (23a). Te new ear trengt model preented in ti paper, a well a te tree ear trengt model adopted by te recent deign guideline are compared wit te collected tet data: a) te Autralian guideline HB 35 (28) wic adopt Cen and Teng (23b) model; b) ACI.44.2R (28) wic adopt Kalia et al. (1998) model; and c) te Italian guideline CNR-DT2 (24) wic adopt Monti and Liotta (27) model. A noted by many reearcer (Cen and Teng 23a, b; Kalia et al. 1998; Monti and Liotta 27; Triantaillou 1998; Triantaillou and Antonopoulo 2), te ear crack angle a a igniicant eect on te FRP ear capacity [alo ee Eq. (5)]. Given te igniicant eect o te ear crack angle, te evaluation preented below are in two tep. In tep one, only toe pecimen wit te experimental ear crack angle inormation (picture, ketce or text decription) in te original ource (wic include 74 pecimen: 46 wit FRP ide trip and 28 wit FRR U-trip) are ued to evaluate te perormance o te model mentioned above. Te reult are own in Fig. 1(a)-1(d) and Fig. 11(a)-11(d) or te propoed model and te model in HB 35 (28), ACI.44.2R (28) and CNR-DT2 (24), repectively. Te coeicient o determination (R 2 ) i own in te repective igure or eac cae. Oter tatitical caracteritic are preented in Table 2. It ould be noted tat or eac o tee ear trengt model, two comparion are made: one wit te eect o ear interaction neglected (Fig. 1), and te oter wit te eect o ear interaction included (Fig. 11). It ould alo be noted tat tee comparion are made between te predictive model and te tet reult, o all partial aety actor or deign ue ave not been included. Figure 1 clearly ow tat i te eect o ear interaction i not conidered, te perormance o te propoed model in predicting experimental reult i imilar to tat o te model in HB 35 (28). Bot model provide igniicantly better prediction tan te oter two model in term o te coeicient o determination (R 2 ) and oter tatitical meaure (ee Table 2). In particular, it i o interet to note tat tree o te our model [i.e. except te CNR-DT2 (24) model] predict an average value o te predicted-to-experimental ratio (reerred to a te average ratio ereater) quite cloe to 9

10 1.. Te ame concluion can be drawn i te prediction or FRP ide trip and FRP U-trip are eparately aeed (Table 2). Te prediction o te CNR-DT2 (24) model ave a very low average ratio o.52 or FRP ide trip but a ig value o 1.26 or FRP U-trip, indicating tat it igniicantly underetimate te FRP ear contribution (V ) or FRP ide trip but overetimate V or FRP U-trip. Te igniicant underetimation o V or FRP ide trip i ciely due to te neglect o te FRP bond lengt above te crack tip and tat below te crack end o te eective ear crack a explained in detail by Cen (21). For FRP U-trip, te overetimation o V i caued by te ollowing reaon according to analye detailed in Cen (21). Firt, unlike te HB 35 (28) model and te propoed model, te CNR-DT2 (24) model doe not peciy an upper bound to te FRP maximum tre. A a reult, te FRP ear contribution may be overetimated i te FRP material a a very low trengt [e.g. or pecimen IIGu in Malek and Saadatmane (1998)]. Second, te model ue.9d to determine te FRP area contributing to te ear capacity (were d i te eective dept o te RC beam) regardle o te actual bond lengt o FRP. Ti may overetimate te FRP ear contribution i te FRP U-trip are bonded to only part o te beam eigt [e.g. pecimen in Kalia and Nanni (2)]. Tird and more importantly, te expreion or te FRP ear contribution in CNR-DT2 (24) i te FRP orce in te iber direction and it i only valid wen te iber orientation i vertical ( β = 9 o ); or oter iber orientation, te expreion i incorrect and may lead to unae reult [e.g. pecimen o Hutcinon and Rizkalla (1999)]. Fig. 11 ow tat i te eect o ear interaction i conidered, te propoed model and te HB 35 (28) model alo perorm better tan te two oter model in term o R 2 ; imilar concluion can be drawn i FRP ide trip and FRP U-trip are aeed eparately a own in Cen (21). I oter tatitical indexe are aeed (Table 2), te Cen and Teng (23b) model in HB 35 (28) i ligtly more accurate in predicting experimental obervation tan te propoed model in term o te coeicient o variation. Bot model provide better prediction tan te oter two model [ACI.44.2R (28) and CNR-DT2 (24)]. By comparing Fig. 1 wit wit Fig. 11, it i clear tat conidering te eect o ear interaction igniicantly improve te perormance o all our model. Te ame concluion can be drawn rom te tatitical indexe in Table 2. In tep two, te ear crack angle i et to be 45 o or all pecimen in te databae a i done in mot deign guideline [e.g. ACI 44.2R (28)]. Te comparion on te bai o ti aumption are own in Fig. 12 and 13, wit te correponding tatitical inormation own in bracket in Table 2. Te aumption lead to muc more conervative prediction or all te model a expected (ee Fig. 12 and 13 and Table 2). Again, it can be een rom te tatitic in Table 2 tat te propoed model and Cen and Teng (23b) model provide te bet prediction wit te propoed model being ligtly better, and tat conidering te eect o ear interaction igniicantly improve te perormance o all our model. DESIGN RECOMMENDATION From te above aement, it can be een tat altoug te propoed model i developed upon a more rigorou bai and provide te bet perormance in predicting te ear contribution o FRP to te ear reitance o te beam wen te ear crack angle i et to 45 o, Cen and Teng (23b) model a adopted in HB 35 (28) i till capable o providing atiactory prediction, particularly wen te eect o ear 1

11 interaction i conidered uing te ear interaction actor (K max ) propoed in ti tudy. In addition, Cen and Teng (23b) model a a impler orm compared wit te propoed model; tu, it i more uitable or deign ue. For FRP ide trip, Eq. (3)-(32) can be employed to conider te eect o ear interaction. For FRP U-trip, in general, Eq. (28) and (29) can be employed to conider te eect o ear interaction. However, te eect o ear interaction i limited i w ep, i larger tan a certain value, and can be neglected. For example, wen wep, 3. [wic eventually lead to a limitation on λ =, e Le according to Eq. (15) or a certain FRP coniguration (i.e. w, and β ) and certain value o inerred rom Eq. (26) tat ' c,θ ), it can be K max i uually larger tan.9 a K i normally larger K = in ti cae, ubject tan.9 (ee Fig. 6 and 7 or reerence) and 1 to μ = V V 1. CONCLUDING REMARKS Built upon te autor previou work a preented in Cen et al. (21) and Cen et al. (211), ti paper a preented a ear trengt model or te FRP debonding ailure mode or RC beam ear-trengtened wit FRP trip. A alient eature o te new model i tat it take into account te proce o debonding ailure (ee Cen et al. 211) and te eect o ear interaction between externally bonded FRP trip and internal teel tirrup (Cen et al. 21). Te new model a been own to perorm well in predicting te ear contribution o FRP by comparing it prediction wit a large tet databae. Perormance comparion between te new ear trengt model and tree oter ear trengt model adopted in exiting deign guideline ave alo been undertaken. Tee comparion indicate tat te new model a te bet perormance among te our model examined, and te incluion o te eect o ear interaction lead to a igniicant improvement to te perormance o all our model. Te reult ave alo revealed tat te model in ACI.44.2R (28) ow unatiactory perormance probably due to it empirical nature and te ue o an inappropriate model or te eective FRP bond lengt; te model in CNR-DT2 (24) generally provide conervative prediction or FRP ide trip but overetimate te ear reitance oered by FRP U-trip. A deign recoendation a been propoed baed on tee comparion. It ould be noted tat te new ear trengt model i baed on two aumption: (a) te FRP debonding ailure proce i dominated by te widening o a ingle critical ear crack; (b) te critical ear crack governing te FRP debonding proce a a linear crack ape. In real RC beam ear-trengtened wit FRP trip, econdary ear crack may exit, and tey can ave a igniicant eect. Te actual widt variation o te critical ear crack i complex and depend on many actor including te amount o teel and FRP ear reinorcement and teel tenion reinorcement. Te eect o tee two aumption ould be examined in uture reearc. ACKNOWLEDGEMENTS Te autor are grateul or te inancial upport received rom te Reearc Grant Council o te Hong Kong Special Adminitrative Region (Project No: PolyU 5151/3E), 11

12 te Nice Area Funding Sceme o Te Hong Kong Polytecnic Univerity, and National Natural Science Foundation o Cina (Project No: ). Tey would alo like to acknowledge te upport rom te Scotti Funding Council or te Joint Reearc Intitute between te Univerity o Edinburg and Heriot-Watt Univerity wic orm part o te Edinburg Reearc Partnerip in Engineering and Matematic (ERPem). Te autor are alo grateul or te ollowing reearcer or teir valuable elp in etabliing te databae preented in ti paper: Dr J.A.O. Barro, L. De Lorenzi, A.J. Beber, G. Kim, H. O, G. Ga and Proeor Z.F. Cen. 12

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14 eectivene wit CFF trip." Eng. Struct., 25(4), Dia, S. J. E., and Barro, J. A. O. (21). "Perormance o reinorced concrete T beam trengtened in ear wit NSM CFRP laminate." Eng. Struct., 32(2), Feng, X. S., Li, J., and Cen, Z. F. (24). "Experimental reearc on ear trengtening o reinorced concrete beam wit externally bonded CFRP eet." Proceeding o te Tird International Conerence on Eartquake Engineering - New Frontier and Reearc Tranormation, W. Q. Liu, F. G. Yuan, and P. C. Cang, ed., Intellectual Property Publ Houe & Cina Waterpower Pre, Beijing, ib (21). "Externally Bonded FRP Reinorcement or RC Structure." Federation Internationale du Beton (ib), Lauanne, Switzerland. Grande, E., Imbimbo, M., and Raulo, A. (29). "Eect o Tranvere Steel on te Repone o RC Beam Strengtened in Sear by FRP: Experimental Study." J. Compo. Contr., 13(5), HB35 (28). "Deign Handbook or RC Structure Retroitted wit FRP and Metal Plate: Beam and Slab." Standard Autralia, GPO Box 476, Sydney, NSW, Autralia. Hollaway, L. C., and Teng, J. G. (28). "Strengtening and Reabilitation o Civil Inratructure Uing Fibre-reinorced Polymer (FRP) Compoite." Woodead Publiing Limited, Cambridge England. Hutcinon, R. L., and Rizkalla, S. H. (1999). "Sear trengtening o AASHTO bridge girder uing carbon ibre reinorced polymer eet." Proc., Te Fourt International Sympoium on Fibre Reinorced Polymer Reinorcement or Reinorcement Concrete Structure ACI Publication SP-188, Kaclakev, D. I., and Barne, W. A. (1999). "Flexural and ear perormance o concrete beam trengtened wit ibre reinorced polymer laminate." Proc., Te Fourt International Sympoium on Fibre Reinorced Polymer Reinorcement or Reinorced Concrete Structure, ACI Publication SP-188, Kage, T., Abe, M., Lee, H. S., and Tomoawa, F. (1997). "Eect o CFRP eet on ear trengtening o RC beam damaged by corroion o tirrup." Proc., Tird International Sympoium on Non-Metallic (FRP) Reinorcement or Concrete Structure, Kalia, A., Gold, W. J., Nanni, A., and Abel-Aziz, M. I. (1998). "Contribution o externally bonded FRP to ear capacity o RC lexural member." J. Compo. Contr., 2(4), Kalia, A., and Nanni, A. (2). "Improving ear capacity o exiting RC T-ection beam uing CFRP compoite." Cem. Concr. Compo., 22(3), Kalia, A., and Nanni, A. (22). "Reabilitation o rectangular imply upported RC beam wit ear deiciencie uing CFRP compoite." Contr. Build. Mater., 16(3), Kalia, A., Tumialan, G., Nanni, A., and Belarbi, A. (1999). "Sear trengtening o continuou reinorced concrete beam uing externally bonded carbon ibre reinorced polymer eet." Proc., Te Fourt International Sympoium on Fibre Reinorced Polymer Reinorcement or Reinorcement Concrete Structure., ACI Publication SP-188, Kim, G., Sim, J., and O, H. (28). "Sear trengt o trengtened RC beam wit FRP in ear." Contr. Build. Mater., 22(6), Li, A., Diagana, C., and Delma, Y. (21). "CRFP contribution to ear capacity o trengtened RC beam." Eng. Struct., 23(1), Li, A., Diagana, C., and Delma, Y. (22). "Sear trengtening eect by bonded compoite abric on RC beam." Compo. Pt. B-Eng., 33(3), Malek, A. M., and Saadatmane, H. (1998). "Analytical tudy o reinorced concrete beam trengtened wit web-bonded iber reinorced platic plate or abric." ACI 14

15 Struct. J., 95(3), Matty, S. (2). "Structural Beaviour and Deign o Concrete Member Strengtened wit Externally Bonded FRP Reinorcement." PD, Univerity o Gent, Belgium, Gent. Mitui, Y., Murakami, K., Takeda, K., and Sakai, H. (1998). "A tudy on ear reinorcement o reinorced concrete beam externally bonded wit carbon iber eet." Compo. Interace, 5(4), Moidi, A., and Caallal, O. (211). "Sear Strengtening o RC Beam wit EB FRP: Inluencing Factor and Conceptual Debonding Model." J. Compo. Contr., 15(1), Monti, G., and Liotta, M. (27). "Tet and deign equation or FRP-trengtening in ear." Contr. Build. Mater., 21(4), Oeler, D. J., and Seracino, R. (24). "Deign o FRP and Steel Plated RC tructure: Retroitting Beam and Slab or Strengt, Stine and Ductility." Elevier, UK. Park, S. Y., Namaan, A. E., Lopez, M. M., and Till, R. D. (21). "Sear trengtening eect o RC beam uing glued CFRP eet." Proc., Te International Conerence on FRP Compoite in Civil Engineering, Pellegrino, C., and Modena, C. (22). "Fiber reinorced polymer ear trengtening o reinorced concrete beam wit tranvere teel reinorcement." J. Compo. Contr., 6(2), Pellegrino, C., and Modena, C. (26). "Fiber-reinorced polymer ear trengtening o reinorced concrete beam: Experimental tudy and analytical modeling." ACI Struct. J., 13(5), Pellegrino, C., and Modena, C. (28). "An experimentally baed analytical model or te ear capacity o FRP-trengtened reinorced concrete beam." Mec. Compo. Mater., 44(3), Rizzo, A., and De Lorenzi, L. (29). "Beavior and capacity o RC beam trengtened in ear wit NSM FRP reinorcement." Contr. Build. Mater., 23(4), Sato, Y., Ueda, T., Kakuta, Y., and Ono, S. (1997). "Ultimate ear capacity o reinorced concrete beam wit carbon ibre eet." Proc., Tird Sympoium on Non-Metallic (FRP) Reinorcement or Concrete Structure, Sato, Y., Ueda, T., Kakuta, Y., and Tanaka, T. (1996). "Sear reinorcing eect o carbon ibre eet attaced to ide o reinorced concrete beam." Proc., Advanced Compoite Material in Bridge and Structure, Sundarraja, M. C., Rajamoan, S., and Bakar, D. (28). "Sear trengtening o RC beam uing GFRP vertical trip - An experimental tudy." J. Rein. Plat. Compo., 27(14), Taljten, B. (23). "Strengtening concrete beam or ear wit CFRP eet." Contr. Build. Mater., 17(1), Taljten, B., and Elgren, L. (2). "Strengtening concrete beam or ear uing CFRP-material: evaluation o dierent application metod." Compo. Pt. B-Eng., 31(2), Teng, J. G., and Cen, J. F. (29). "Mecanic o debonding in FRP-plated RC beam." Proc. Int. Civil Eng.-Struct. Build., 162(5), Teng, J. G., Cen, J. F., Smit, S. T., and Lam, L. (22). "FRP-Strengtened RC Structure." Jon Wiley and Son, Ciceter, Wet Suex, UK. Teng, J. G., Lam, L., and Cen, J. F. (24). "Sear trengtening o RC beam uing FRP compoite." Progre in Structural Engineering and Material, 6(3), Triantaillou, T. C. (1998). "Sear trengtening o reinorced concrete beam uing 15

16 epoxy-bonded FRP compoite." ACI Struct. J., 95(2), Triantaillou, T. C., and Antonopoulo, C. P. (2). "Deign o concrete lexural member trengtened in ear wit FRP." J. Compo. Contr., 4(4), Uji, K. (1992). "Improving ear capacity o exiting reinorced concrete member by applying carbon ibre eet." Tranaction o te Japan Concete Intitute, 14, Zang, Z. C., and Hu, C. T. T. (25). "Sear trengtening o reinorced concrete beam uing carbon-iber-reinorced polymer laminate." J. Compo. Contr., 9(2),

17 LIST OF TABLES Table 1. Experimental Databae or FRP Sear-Strengtened RC beam Failing by FRP Debonding Table 2. Perormance o Sear Strengt Model (reult or a 45 o ear crack angle are lited in bracket) 17

18 LIST OF FIGURES Fig. 1. Development o dierent component o ear reitance wit te crack end widt o te critical ear crack Fig. 2. Notation or a general ear trengtening ceme Fig. 3. Example FRP ear contribution veru crack end widt Fig. 4. Development o FRP mobilization actor Fig. 5. Notation or te ultimate tate o ear-trengtening FRP trip (Cen et al. 211):(a) FRP ide trip; (b) FRP U-trip Fig. 6. Comparion between propoed expreion or K and numerical prediction or plain bar wit φ = 8 Fig. 7. Comparion between propoed expreion or K and numerical prediction or deormed bar wit φ = 1 Fig. 8. Development o ear interaction actor K wit crack end widt w e ' ( E = 23 GPa, t =.11, = 4, = 3 MPa ; plain bar wit, e c φ = 8, = 3 MPa ): (a) FRP ide trip; (b) FRP U-trip y Fig. 9. Comparion o maximum ear interaction actor between prediction o Eq. (3) and reult rom accurate analyi ( K max,a ) or dierent λ =, e Le : (a) plain bar ( φ = 8, = 35 MPa ); (b) deormed bar ( φ = 8, = 55 MPa ) y Fig. 1. Predicted veru experimental FRP ear contribution (ear interaction eect ignored) or pecimen wit experimental crack angle: (a) propoed model; (b) HB 35 (28); (c) ACI 44.2R (28); (e) CNR-DT2 (24) Fig. 11. Predicted veru experimental FRP ear contribution (ear interaction eect included) or pecimen wit experimental crack angle: (a) propoed model; (b) HB 35 (28); (c) ACI 44.2R (28); (e) CNR-DT2 (24) Fig. 12. Predicted veru experimental FRP ear contribution (ear interaction eect ignored) or a 45 o ear crack angle: (a) propoed model; (b) HB 35 (28); (c) ACI 44.2R (28); (d) CNR-DT2 (24) Fig. 13. Predicted veru experimental FRP ear contribution (ear interaction eect included) or a 45 o ear crack angle: (a) propoed model; (b) HB 35 (28); (c) ACI 44.2R (28); (d) CNR-DT2 (24) y 18

19 Table 1. Experimental Databae or FRP Sear-Strengtened RC beam Failing by FRP Debonding Reerence Speci. Beam propertie FRP propertie Steel tirrup propertie Tet reult c MPa b w d Sect. Type Con. E GPa t MPa w S β deg. Type Ф S E GPa y MPa θ deg. V u,tet kn V,tet kn Uji (1992) R C SP NA R C SP NA R C SP NA Al-Sulaimani SO R C SS D et al. (1994) SP R C SS D WO R C SP D WP R C SP D Sato et al. S R C SS NA (1996) S R C SP NA Kage et al. SB R C SP R (1997) SB R C SP R SB R C SP R SB R C SP R Caallal et al. RS R C SS D (1998) RS R C SS D Mitui et al. (1998) Triantaillou (1998) Kaclakev and Barne (1999) A R C SP NA B R C SP NA C R C SP NA D R C SP NA E R C SP NA S1a R C SS NA S1b R C SS NA S1(45) R C SS NA S2a R C SS NA S2b R C SS NA S2(45) R C SS NA S3a R C SS NA S3b R C SS NA S3(45) R C SS NA C1, 2L R C SP NA C1, 3L R C SP NA C2, 3L R C SP NA SR R C SS NA Taljten and Elgren(2) SR R C SP NA Re.[1] BT T C SS NA

20 Table 1. (Continued.) Reerence Park et al. (21) Pellegrino and Modena (22) Taljten (23) Beber and Campo Filo (25) Zang and Hu (25) Corolin and Taljten (25) Speci. c MPa Beam propertie FRP propertie Steel tirrup propertie Tet reult b w d Sect. Type Con. E GPa t MPa w S β deg. Type Ф R C SS NA T C SS D TR3C R C SP D TR3C R C SP D TR3C R C SP D TR3D R C SP D TR3D R C SP D TR3D R C SP D TR3D R C SP D TR3D R C SP D RC R C SP NA C R C SP NA C R C SP NA C R C SP NA C R C SP NA V9_A R C SS NA V9_B R C SS NA V21_A R C SS NA V12_B R C SS NA V14_B R C SS NA V13_A R C SP NA V13_B R C SP NA V14_A R C SP NA V15_A R C SP NA V2_B R C SS NA V22_B R C SS NA V21_B R C SS NA V22_A R C SS NA Z R C SS NA Z R C SS NA Z R C SS NA a R C SP NA b R C SP NA R C SP NA R C SP D S E GPa y MPa θ deg. V u,tet kn V,tet kn 2

21 Table 1. (Continued.) Reerence Speci. Beam propertie FRP propertie Steel tirrup propertie Tet reult c MPa b w d Sect. Type Con. E GPa t MPa w S β deg. Type Ф S E GPa y MPa θ deg. V u,tet kn V,tet kn Re.[2] R C SP D Re.[3] V R G SS NA Kim et al. (28) CP2-1D R C SS NA Grande and Raulo (29) CP3-1V R C SS NA RS2Sa R C SP D RS3Sa R C SP D RS3Sb R C SP D RS4Sb R C SP D Sato et al. S R C US NA (1996) S R C UP NA Re.[4] No T C UP R Re.[5] IIGu R C UP R Kalia et al. CO R C US NA (1999) CO R C UP NA Re.[6] S-Di-CL I C US D Kalia and BT T C UP NA Nanni BT T C UP NA (2) BT T C US NA Re.[7] BS R C US D Deniaud and Ceng (21) Kallia and Nanni (22) Diagana et al. (23) Adikary et al. (24) T6NS T C US NA T6S T C US R T6S T G UP R SO R C US NA SO R C US NA SO R C US NA SO R C UP NA SO R C UP NA PU R C US R PU R C US R PU R C US R PU R C US R C R C UP NA A R A UP NA

22 Table 1. (Continued.) Reerence Feng et al. (24) Beber and Campo Filo (25) Pellegrino and Modena (26) Speci. c MPa Beam propertie FRP propertie Steel tirrup propertie Tet reult b w d Sect. Type Con. E GPa t MPa w S β deg. Type Ф SB R C UP R SB R C UP R SB R C US R SB R C US R SB R C US R SB R C US R SB R C US R SB R C US R V1_A R C US NA V1_B R C US NA V17_A R C US NA V11_A R C US NA V11_B R C US NA V17_B R C US NA V19_A R C US NA V19_B R C US NA V15_B R C UP NA V16_B R C UP NA A-U1-C R C UP D A-U2-C R C UP D A-U1-C R C UP D A-U2-C R C UP D A-U1-S R C UP D A-U1-S R C UP D Re.[8] A12_M R C US NA Re.[9] UW R C UP D Dia and 2S-M(1) T C US D Barro(21) 2S-M(2) T C US D Note: Re.[1]=Kalia and Nanni (2); Re.[2]=Carolin and Taljten (25); Re.[3]=Sundarraja et al. (28); Re.[4]=Sato et al. (1997); Re.[5]=Malek and Saadatmane (1998); Re.[6]=Hutcinon and Rizkalla (1999); Re.[7]=Matty (2); Re.[8]= Barro and Dia (26); Re.[9]=Rizzo and Lorenzi (29); Section o beam: R=Rectangular ection, T=T-ection, b w =web widt o beam, =eigt o beam, d=eective dept o beam; FRP material: C=CFRP, G=GFRP, A=AFRP, E =elatic modulu o FRP, t =nominal tickne o FRP, =trengt o FRP, w =widt o FRP in direction perpendicular to iber direction, =center-to-center pacing o FRP trip in te beam axial direction, β=iber orientation wit repect to beam axi; Speci.=Specimen; Sect.=Section o beam; Con.=FRP Coniguration; deg.=degree; SSxx=ide trip wit β=xx o, SPxx=ide eet wit β=xx o, USxx=U-trip wit β=xx o, UPxx=U-eet wit β=xx o ; Steel tirrup: R=round bar, D=deormed bar, Ф =diameter, =pacing, E =elatic modulu; y =yield trengt; c =cylinder compreive trengt o concrete; θ=angle o critical ear crack oberved in tet [aumed to be 45 o i tere i no tet inormation except toe pecimen in te databae o Cen and Teng (23b) or wic te crack angle were taken rom Cen and Teng (23b)]; V u,tet =tet ear capacity o beam; V,tet =tet ear contribution o FRP. S E GPa y MPa θ deg. V u,tet kn V,tet kn 22