NSM FRP STRIPS SHEAR STRENGTH CONTRIBUTION TO A RC BEAM: A DESIGN PROCEDURE Vincenzo Bianco, Giorgio Monti and J.A.O. Barros

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1 NSM FRP STRIPS SHEAR STRENGTH CONTRIBUTION TO A RC BEAM: A DESIGN PROCEDURE inenzo Biano, Giorgio Monti and J.A.O. Barros Synopsis: This paper presents a losed-orm proedure to evaluate the shear strength ontribution provided to a Reinored Conrete (RC beam by a system o Near Surae Mounted (NSM Fiber Reinored Polymer (FRP strips. This proedure is based on the evaluation o: a the onstitutive law o the average-available-bond-length NSM FRP strip eetively rossing the shear rak and b the imum eetive apaity it an attain during the loading proess o the strengthened beam. Due to omplex phenomena, suh as: a interation between ores tranerred through bond to the surrounding onrete and onrete rature, and b interation among adjaent strips, the NSM FRP strip onstitutive law is largely dierent than the linear elasti one haraterizing the FRP behavior in tension. One the onstitutive law o the average-available-bond-length NSM strip is reliably known, its imum eetive apaity an be determined by imposing a oherent kinemati mehanism. The sel-ontained and ready-to-implement set o analytial equations and logial operations is presented along with the main underlying physial-mehanial priniples and assumptions. The ormulation proposed is appraised against some o the most reent experimental results and its preditions are also ompared with those obtained by a reently developed more sophistiated model. Keywords: Conrete Frature; Debonding; Design; FRP; NSM; Shear Strengthening; Tensile Rupture.

2 0 0 0 inenzo Biano is a Post Do at the Department o Strutural Engineering and Geotehnis o the Sapienza University o Rome, Italy. He reeived his PhD rom the Sapienza University o Rome. His researh interests inlude seismi assessment and retroit o existing strutures, mehanial modeling and use o omposite materials or strutural rehabilitation. Giorgio Monti is a Full Proessor at the Department o Strutural Engineering and Geotehnis o the Sapienza University o Rome, Italy. He reeived is MS rom the University o Berkley, Caliornia, and his PhD rom the Sapienza University o Rome. His Researh interests span rom reliability to the assessment and retroitting o existing strutures in seismi zones. Joaquim Barros is an Assoiate Proessor with aggregation and Diretor o the aboratory o the Strutural Group o the Department o Civil Engineering, University o Minho at Guimarães, Portugal. He reeived his MS and PhD rom the University o Porto, Portugal. He is a member o ACI Committees 0 and, and ib TG. and.. His researh interests inlude strutural strengthening, omposite materials, iber reinored onrete and inite element method. INTRODUCTION Shear strengthening o RC beams by NSM tehnique onsists o bonding FRP strips by a powerul strutural adhesive into thin shallow slits ut onto the onrete over o the beam web lateral aes. A omprehensive three-dimensional mehanial model to predit the NSM FRP strips shear strength ontribution to a RC beam was reently developed (Biano 00, Biano et al. 00a-b and 0. Despite its onsisteny with experimental reordings, that model turned out to be somehow umbersome to be easily implemented and aepted by proessional strutural engineers. The aim o the present work is to develop a simpler omputational proedure that has to be: a mehanially-based and b simple to implement. As to the irst point, it has to ulill equilibrium, kinemati ompatibility and onstitutive laws. As to the seond point, it has to be a design tool easy to apply. For this purpose, a reasonable ompromise between auray o predition and omputational demand has to be ahieved. Exessively simpliied assumptions, whih would provide too roughly onservative estimates o the shear strength ontribution provided by a system o NSM FRPs, should be avoided sine they ould lead to uneonomial design solutions, disouraging appliation, urther improvement and spreading o the tehnique. A relatively simple model an be derived rom the more sophistiated one by introduing the ollowing simpliiations (Biano 00: a bi-linear rigid-sotening loal bond stress-slip diagram is adopted instead o a multi-linear diagram, onrete rature surae is assumed as semi-pyramidal instead o semionial, attention is oused on the average-available-bond-length NSM FRP strip glued on the relevant prism o surrounding onrete, determining the onstitutive law o the average-available-bond-length NSM strip, along the approah ollowed or Externally Bonded Reinorement (EBR by Monti et al. (00, and determining the imum eetive apaity attainable by the average-available-bond-length NSM strip plaed along the CDC, imposing a oherent kinemati mehanism (e.g. Monti et al. 00, Monti and otta 00. The main eatures o the resulting modeling strategy are reported hereater.

3 Beam lange Z ξ CDC plane d O θ β x i web intrados web extrados a i ( t n γ ( t n s onrete rature surae X E FRP strip B h w l X i s t B τ τ 0 sotening rition 0.0 i tr, i b ree slipping view A-A FRP strip A α b w CDC plane FRP strip α α θ β s A θ l Oi β d s b w CDC plane α view B-B Debonding Strip tensile rupture Conrete semipyramidal rature e Mixed shallowsemi-pyramid-plusdebonding Figure Main physial-mehanial eatures o the alulation proedure: a average-available-bond-length NSM strip and relevant prism o surrounding onrete, b adopted loal bond stress-slip relationship, NSM strip onined to the orresponding onrete prism o surrounding onrete and semi-pyramidal rature surae, d setions o the onrete prism. During the loading proess o a RC beam subjet to shear, when onrete average tensile strength tm is attained at the web intrados (Fig., some shear raks originate therein and suessively progress towards the web extrados. Those raks an be thought as a single Critial Diagonal Crak (CDC inlined o an angle θ with respet to the beam longitudinal axis (Fig. a. The CDC an be represented by an inlined plane dividing the web into two portions sewn together by the rossing strips (Fig. a. At load step t, the two web parts, separated by the CDC, start moving apart by pivoting around the rak end whose trae, on the web ae, is point E in Fig. a. From that step on, by inreasing the applied load, the CDC opening

4 angle γ ( t n progressively widens (Fig. a. The strips rossing the CDC oppose its widening by anhoring to the surrounding onrete to whih they traner, by bond, l the ore originating at their intersetion with the CDC, O i, as a result o the imposed end slip [ γ ( t n ]. The apaity o eah strip is provided by its available bond length i that is the shorter between the two parts into whih the rak divides 0 its atual length (Fig. a. Bond is the mehanism through whih stresses are tranerred to the surrounding onrete (Yuan et al. 00, Mohammed Ali et al. 00 and 00, Biano et al. 00. The loal bond stress-slip relationship τ (, omprehensively simulating the mehanial phenomena ourring at the strip-adhesive interae, within the adhesive layer and at the adhesive-onrete interae, an be represented, in a simpliied way, by a bi-linear urve (Fig. b. The subsequent phases undergone by bond during the loading proess, representing the physial phenomena ourring in sequene within the adhesive layer by inreasing the imposed end slip, are: rigid, sotening rition and ree slipping (Fig. b (Biano 00. The onstitutive law i ( Ri ; o an NSM FRP strip, i.e. the ore transmissible by a strip with resisting bond length Ri as untion o the imposed end slip, an be determined by analyzing the behavior o the simple strutural element omposed o the NSM FRP strip within a onrete prism (Fig. a,-d whose transversal dimensions are limited by the spaing s between adjaent strips and hal o the web ross setion width b w. In this way, the problem o interation between adjaent strips (e.g.: Dias and Barros 00, Rizzo and De orenzis 00 is taken into aount in a simpliied way, i.e., by limiting the onrete volume into whih subsequent ratures an orm, to the amount o surrounding onrete pertaining to the single strip in dependene o s and b w. Moreover, even though here negleted, the interation with existing stirrups may be also aounted or by limiting the transversal dimension o the onrete prism to a ertain ratio o b w, 0 0 sine the larger the amount o stirrups, the shallower onrete rature is expeted to be (Biano et. al 00 even i, in this respet, urther researh is neessary. In partiular, in the present work, attention is oused on the system omposed o the strip with the average value o available bond length glued on the pertaining prism o surrounding onrete (Fig. -d. The ailure modes o an NSM FRP strip subjet to an imposed end slip omprise, depending on the relative mehanial and geometrial properties o the materials involved: debonding, tensile rupture o the strip, onrete semi-pyramidal tensile rature and a mixed shallow-semi-pyramid-plus-debonding ailure mode (Fig. e. The term debonding is adopted to designate loss o bond due to damage initiation and propagation within the adhesive layer and at the FRP strip-adhesive and adhesiveonrete interaes, so that the strip pulling out results (Fig. e. When prinipal tensile stresses tranerred to the surrounding onrete attain its tensile strength, onrete ratures along a surae, envelope o the ompression isostatis, whose shape an be onveniently assumed as a semi-pyramid with prinipal generatries inlined o an angle α with respet to the strip longitudinal axis (Fig. -d. Inreasing the imposed end slip an result in subsequent semi-pyramidal and oaxial rature suraes in the onrete surrounding the NSM strip. These progressively

5 redue the resisting bond length Ri that is the portion o the initial available bond length i still bonded to onrete. Those subsequent ratures an either progress up to the ree end, resulting in a onrete semi-pyramidal ailure, or stop progressing midway between loaded and ree end, resulting in a mixed-shallow-semi-pyramid-plus-debonding ailure (Fig. e. Moreover, regardless o an initial onrete rature, the strip an rupture (Fig. e. The ormulation obtained by this strategy is presented in the ollowing setions along with the main mehanial bases. RESEARCH SIGNIFICANCE A alulation proedure was developed to evaluate the NSM FRP strips shear strength ontribution to a RC beam. The equations and the logial operations neessary to implement the proposed proedure are presented along with the theoretial bases orm whih they originate. CACUATION PROCEDURE The input parameters inlude (Figs. -: beam ross-setion web s depth h w and width b w ; inlination angle o both CDC and strips with respet to the beam longitudinal axis, θ and β, respetively; strips spaing measured along the beam 0 axis s ; angle α between axis and prinipal generatries o the semi-pyramidal strength 0 rature surae (Fig. -d; onrete average ompressive strength m ; strips tensile u and Young s modulus E ; thikness a and width b o the strip ross-setion; inrement ɺ o the imposed end slip; values o bond stress τ 0 and slip deining the adopted loal bond stress-slip relationship (Fig. b: τ 0 0 τ ( < = ( 0 > The geometrial oniguration is adopted in whih the minimum integer number l,int N o strips ross the CDC with the irst one plaed at a distane equal to s rom the rak origin (Fig. a. This oniguration orresponds to the minimum o the sum o all the available bond lengths i. N,int is obtained by rounding o the real number to the lowest integer, as ollows: l ( otθ + ot β N,int = round o hw ( s and the average available bond length i is obtained by: i with: = ( N l,int l N,int i= i l

6 sinθ hw i s or x i < (ot θ + ot β sin( i = θ + β sinθ hw i s or x i (ot θ + ot β sin( θ + β ( and: x i = i s ( Ater having deined the geometrial harateristis o the simple strutural system omposed o the average-available-bond-length strip within the relevant prism o surrounding onrete, it is neessary to determine its onstitutive law i ( ; and the orresponding imum eetive apaity One i has been obtained, the atual and design shear strength ontribution an be obtained by Eq. (. Input Parameters h ; b ; α; ; s ; β; ; E ; a ; b ; τ ; ; ɺ ; θ w w m u 0 i, as explained hereater. d values o the NSM Evaluation o the average value o the available bond length and the minimum number o strips rossing the CDC = h s N l ( ; θ; β; ;,int i w Evaluation o the bond onstitutive law o the average length NSM strip ( ; = ( τ 0; ; ; ; ; ; ; i i m w a b E s b Evaluation o the omprehensive onstitutive law o the average length strip i ( ; = i ( ; ; m; s ; bw ; u; α Evaluation o the imum eetive apaity o the average NSM FRP strip i, e Appliation o the Shear Formula = N sin β l,int i, e Figure Calulation proedure: main algorithm. CONSTITUTIE AW OF A SINGE NSM FRP STRIP The simple strutural system omposed o a single strip, the adhesive and the surrounding onrete, undergoes hanges during the loading proess sine, eah time onrete ratures, the resisting bond length redues aordingly. In partiular, the dierent eatures assumed by that system throughout the loading proess are untion not only o the load step t n, but also o the iteration q m in orrespondene o t n (Biano 00. In at, or eah t n, that system undergoes modiiations up to reahing the equilibrium oniguration q e. Whenever onrete ratures, the 0 mehanism o ore traner to the surrounding onrete leaps orward towards the strip s ree end. In general, in orrespondene o eah leap, the overall traner length ( tr, i Ri; inreases and the resisting bond length dereases (Fig. -d. Thus, in general, at eah leap, onrete tensile rature apaity inreases and at the same

7 time the bond-tranerred ore dereases, until equilibrium is attained. In this senario, in order to determine the omprehensive onstitutive law i ( i; o the average-available-bond-length NSM FRP strip bonded to the relevant prism o surrounding onrete, it is neessary to arry out an inremental proedure that simulates the imposed end slip ( t n and to hek, at eah t n, either i onrete is apable o arrying the bond-tranerred stresses without undergoing rature, or i a onrete rature ours and the system has to be modiied aordingly. 0 Bond-based onstitutive law The bond behaviour o an NSM FRP strip subjet to an inreasing imposed end slip an be modelled by ulilling equilibrium, kinemati ompatibility and onstitutive laws o both adhered materials (onrete and FRP and loal bond between themselves (Biano 00. In this way, it is possible to obtain losed-orm analytial equations or both the bond-based onstitutive law i ( Ri ; and the orresponding bond traner length tr, i ( Ri; i ( Ri ; and tr, i ( Ri; length o o a single strip. The latter two quantities,, represent: the ore a strip o resisting bond Ri an traner by bond, as untion o, and the orresponding amount Ri along whih bond is mobilized, respetively. The analytial equations o tr, i ( Ri ; and i ( Ri ;, are presented below and plotted in Fig.. Those analytial equations envisage, or a given ( Ri, three phases, whose limits ; ; are untion o the value assumed by Ri with respet to the eetive bond length tr that is the value o resisting bond length beyond whih any urther inrease o length does not produe any urther inrease o the imum ore transmissible by bond. The bond traner length is as ollows: ( ; ( tr, i Ri tr = = λ aros λ τ 0 J tr, i ( Ri < tr; = Ri, ; = + ( 0.0 Ri s ( Ri < ( Ri tr i ( Ri tr tr tr ( tr, i ( Ri ; = Ri ( Ri < ( Ri tr, i ( Ri ; = 0.0 > ( Ri and the bond-based onstitutive law: i ( Ri ; = p J λ { C os( λ tr ( C sin( λ tr ( } ( < ; = J λ i Ri tr p C x C x i Ri tr = tr ( os( λ sin( λ ( ( ; tr Ri ( 0.0 Ri ( < ( Ri Ri ( (

8 i ( Ri ; = p J λ tr ( Ri < ( Ri C os( λ x C sin( λ x s tr + tr ( Ri i ( Ri ; = 0.0 > ( Ri where: = b + a ( p is the eetive perimeter o the strip ross-setion, and: p A E E A = ; J = + ; J = λ τ 0 J A E A E E A + E A E A E A τ J τ J J = ; C = ; C = A E + A E λ λ ( p 0 0 are bond-modeling onstants (Biano 00, Biano et al. 00b, with A = a b and A = s b the ross-setion o the strip and the onrete prism, respetively. w Moreover, the eetive bond length tr and the orresponding imum bond ore are given by: π τ 0 J tr = ; = p J λ λ λ ( The value o resisting bond length undergoing sotening rition, as untion o the imposed end slip is given by: λ tr ( = aros ( λ τ 0 J and the value o resisting bond length undergoing ree slipping: s A J ( tr ( = ( The resisting bond length-dependent values o imposed end slip deining the extremities o the three bond phases, are given by (Fig. : τ 0 J C sin ( λ Ri + C os( λ Ri + or Ri < tr ( Ri = λ ( or Ri tr or Ri < tr ( Ri = ( + ( Ri tr or Ri tr A J ( Ri = + A Ri Conrete tensile rature apaity J ( The onrete tensile rature apaity ( tr, i i ( is obtained by spreading the onrete average tensile strength tm over the semi-pyramidal surae (Fig. -d o

9 height equal to the total traner length tr, i, orthogonally to it in eah point. By integrating one obtains: bw i ( tr, i = tm min{ tr, i tan α; } sin ( θ + β ( s sin β tr, i sinα s sin β tr, i sinα min ; + min ; sin ( θ + β sin ( θ + β + α sin ( θ + β sin ( θ + β α where tm an be determined rom the average ompressive strength. The total traner length is evaluated as reported in next Eq. (. ( tr, i Ri; ( = ; tr, i Ri tr ( > ; tr, i Ri tr tr ( ; i Ri ( ( ( ( < = = > Ri tr Ri tr Ri tr Ri tr st ( < Ri tr st ( < ; tr, i Ri tr nd nd nd a rd rd ( < Ri tr ( > Ri tr phase 0 st nd rd ( = Ri tr ( > Ri tr < ( Ri ( < ( ( < ( Ri Ri Ri Ri rd ( ; i Ri st st nd nd nd rd b ( > i Ri tr ; rd rd ( = i Ri tr ; ( < i Ri tr ; Ri > Ri tr = Ri tr st < Ri tr with tr eetive bond length Figure Bond-based onstitutive law o a single NSM FRP strip: (a relationship between bond traner length tr, i ( ; Ri values o resisting bond length and imposed end slip or dierent Ri ; (b bi-dimensional and ( three-dimensional

10 representation o the relationship between ore tranerrable by bond i ( ; Ri and or dierent values o Ri. Comprehensive onstitutive law At the n qm : q qe is arried out in order to determine the equilibrium ondition ( q e in the surrounding onrete depending on the urrent value o both imposed end slip ( n and resisting bond length Ri ( tn; q m (Fig.. In partiular, at the m t n load step, based on Ri ( tn; q m and ( n, the bond traner length tr, i Ri ( tn; qm ; ( tn and the orresponding bond-tranerred ore i Ri ( tn; qm ; ( tn are evaluated as reported in Eq. ( and Eq. (, respetively. Then, the urrent value o the total traner length is evaluated as ollows: tr, i ( tn; qm = ( tn ; qe + tr, i Ri ( tn; qm ; ( tn + ( tn; qm ( where ( t ; q is the umulative depth o the onrete rature surae resulting i n e to the urrent q m (Fig. : q m ( ; t q = ( t ; q ; ( t 0 rom the equilibrium o the preeding tn load step and i ( tn; qm is the inrement o onrete rature depth orresponding to the urrent t n, aumulated up i n m tr, i Ri n m n q ( Then, ater having evaluated the onrete rature apaity ( tr, i as indiated in Eq. (, i it is: i Ri ( tn; qm ; ( tn i tr, i ( tn; qm ( meaning that the surrounding onrete is not apable to arry the bond-tranerred ore, then it ratures and the bond traner mehanism leaps orwards towards the ree end. Thus, the parameters Ri ( tn; q m + and i ( tn; q m + are updated ( Ri ( tn; qm+ Ri ( tn; qm tr, i Ri ( tn; qm ; ( tn, i ( tn; qm+ = ( tn; qm + tr, i Ri ( tn; qm ; ( tn and iteration is perormed ( q m + (Fig.. At eah o those leaps, the point representative o the strip state moves rom one bond-based onstitutive law i Ri ( tn; qm ; to the other i Ri ( tn; qm+ ; Ri is larger or equal to the neessary bond traner length tr [ ( tn ], suh leap is only visible in a three dimensional representation (Fig.. The neessary bond traner length tr [ ( tn ] is the bond traner length that would be neessary, i were ininite, to transmit 0 the orresponding ore to the surrounding onrete, with tr [ ( tn ] = tr [ ( tn ] or ( t n and [ ( ] s [ ( ] tr tn = tr + tr tn or ( t n > (Fig.. Note Ri i

11 also that, at eah q m iteration, the equality ( t ; q + ( t ; q + ( t ; q = ( t ; q = has to be ulilled (Fig.. Ri n m i n e i n m Ri i ( ; i Ri i ( t ; q i Ri( tn; q ; n ( tr, i Ri; ( t ; q i n e ( t ; q i n e a ; ( t i i n i i Ri ( tr, i ( ; ( tr, i i ( t q ; i n e ( t q ; i n e [ ( t ] tr, i n ( t tr, i i ; n tr, i ( Ri ; [ ( t ] tr, i n Ri = i NSM strip Ri tr, i onrete rature surae α = 0 tr, i i i i = 0 ( tn ; q ( tn ; qe = q ( tn; q ( tn; qe = q i Ri ( t; q ; ( ; i Ri b tr, i Ri tr, i tr, i Ri i tr, i tr, i ( ; ; i Ri tn qe i Ri ( tn; qe ; Ri Ri ( t; q ( ; Ri tn qe ( t ; q Ri n e d ( ; i i Figure Single NSM FRP strip omprehensive onstitutive law in ase in whih onrete rature remains shallow: a resulting onstitutive law i ( i; dimensional representation, b resulting overall traner length tr, i ( i; in a bi-, setion o the onrete prism and ourrene o subsequent ratures and d resulting

12 in a three-dimensional representation. Note that this plot has been done or an initial resisting bond length equal to the eetive bond length. More in detail, at the q m iteration o the t n load step, i onrete is not in equilibrium ( e = 0, one o the ollowing alternatives might our: onrete rature was deep ( d = but it did not reah the ree end, i.e. the updated resisting bond length Ri ( tn; q m + is not long enough to mobilize, or the urrent ( n, a bond traner length as large as the neessary one: Ri < tr [ ( tn ]. Note that in this ase, the passage o the point representative o the strip state rom one bond-based onstitutive law to the other is also visible in a bi-dimensional representation. Further details an be ound elsewhere (Biano 00; onrete rature was deep ( d = and it reahed the ree end, i.e. the updated resisting bond length Ri ( tn; q m + is null. Note is taken o the urrent value o the imposed end slip ( u ( n and the inremental proedure is terminated sine a deision about the omprehensive onstitutive law an already be taken ( u = (Fig. a. On the ontrary, i at the m t n load step, onrete is in equilibrium 0 ( e =, it is not neessary to iterate and one o the ollowing alternatives might our: the urrent value o bond-tranerred ore is larger or equal to the strip tr tensile rupture apaity ( i. The inremental proedure is terminated sine, even i the surrounding onrete is in equilibrium, the strip has ruptured ( u = and note is taken o the ultimate imposed end slip ( u ( n the next value o the imposed end slip ( n+ is larger or equal to the one in orrespondene o whih the peak bond ore is attained or the urrent value o the resisting bond length ( tn+ Ri ( tn; qe i 0 starts to derease or ( n+, the inremental proedure is terminated and note is taken o the urrent value o the resisting bond length ( Ru Ri and o its relationship with the eetive bond length tr ( u i Ru < tr, u i Ru = tr or u i Ru > tr. onrete rature was deep ( d = and it did not reah the ree extremity. The inremental proedure is terminated ( u = (Fig. d; the next value o the imposed end slip ( n+ is smaller than the one where the peak bond ore is attained or the urrent value o the resisting bond length ( tn+ Ri ( tn; qe. Then, the imposed end slip is inremented and the iteration arried out. onstitutive law i ( i;

13 0 The inremental proedure desribed above is terminated and, depending on the phenomenon haraterizing the speii ase at hand and the type o onstitutive law assoiated ( u, the parameters neessary to deine i ( i; are returned, i.e.: deep onrete rature that reahes the strip s ree extremity ( u = or tensile rupture o the strip ( u =. The parameter neessary to determine the onstitutive law is the imposed end slip u in orrespondene o whih the peak o i ( ; ours. i ( i; is given by the irst bond phase o Eq. ( or 0.0 u (Fig. a; shallow or absent onrete rature with an ultimate value o resisting bond length smaller ( u =, equal ( u = or larger ( u = than the eetive bond length. The parameter neessary to determine the omprehensive onstitutive law is the ultimate value assumed by the resisting bond length Ru i ; is given by Eq. ( or Ri = Ru (Fig. b-;. ( deep tensile rature with an ultimate value o resisting bond length very short but not null ( u =. The parameters neessary to determine the omprehensive onstitutive law are both the imposed end slip in orrespondene o whih the peak o i ( i; value assumed by the resisting bond length u ours and the ultimate Ru. i ( i; is given by: the irst bond phase o Eq. ( or 0.0 u, the seond bond phase o Eq. ( or u ( Ru ( Ru ( Ru < and the third bond phase o Eq. ( or < (Fig. d. ( ; ( ; i Ri u = or u = u i Ri u = = Ru tr u ( ; i Ri u = < tr a i ( Ri; b u = u > = ; Ru tr Ri u Ru 0 d Figure Possible omprehensive onstitutive law o a NSM FRP strip onined to a prism o onrete: (a onrete that reahes the ree extremity ( u = or strip tensile rupture ( u =, superiial and/or absent onrete rature and ultimate resisting bond length (b smaller ( u = or equal ( u = or ( larger ( u = than the eetive bond length and (d deep onrete rature ( u =. u

14 MAXIMUM EFFECTIE CAPACITY OF A SINGE NSM FRP STRIP The eetive apaity ( γ is the average o the NSM FRP strip apaity along i, e the CDC ( γ ξ or a given value o the CDC opening angle γ (e.g. Fig., i, CDC ; where ξ is the reerene system assumed along the CDC (Fig. a. ( γ ξ is i, CDC ; obtained by introduing the kinemati ompatibility ( ( γ ; ξ = ξ γ sin ( θ + β into the omprehensive onstitutive law o the single average-available-bond-length NSM FRP strip i ( ;. For the sake o brevity, all o the details are herein omitted but they an be ound elsewhere (Biano 00. The equation to evaluate the imum eetive apaity i, e and the value o the CDC opening angle γ in orrespondene o whih it is attained, assume dierent eatures as untion o the type (u o the omprehensive onstitutive law haraterizing the speii ase at hand. ( γ ξ i, CDC ; γ =. γ ( ; i Ru u < γ 0. γ ξ = 0 ( γ ; ξ a ξ = d i, e = ( γ u b γ =. γ = ( γ i, e i, e 0 = u γ γ 0. γ ξ = 0 ξ = d γ γ γ d Figure Maximum eetive apaity along the CDC or the ases o onrete rature that reahes the strip s ree extremity ( u = or strip s tensile rupture ( u = : a apaity i, CDC ( γ ; ξ and imposed end slip, CDC ( γ ; ξ distribution along the CDC or dierent values o the CDC opening angle γ, b omprehensive onstitutive law and d eetive apaity as untion o the CDC opening angle γ. Cases o onrete rature that reahes the strip s ree extremity (u = or strip tensile rupture (u =

15 0 In these ases, the exat value o the imum eetive apaity is attained or a value o the CDC opening angle γ suh as to yield an imposed end slip at the end o the rak ( ( d, equal to u (Fig. i.e.: A C i, e = i, e ( γ = A C d γ + d A γ π arsin( A γ d ( A γ d ( A γ (0 + d } where: p J λ sin( θ + β λ sin( θ + β A = ; A = p J λ ; A = τ 0 J τ 0 J ( are integration onstants independent o the type (u o omprehensive onstitutive law and: u γ = γ = sin ( θ + β ( d Case o shallow onrete rature and strip ultimate resisting bond length smaller than the eetive bond length (u = In this ase, the imum eetive apaity is attained or a value o γ very lose to γ that is the value o the CDC opening angle suh as to yield an imposed end slip at the end o the rak, equal to ( Ru. For the sake o simpliity, it is assumed that i is eetively attained or γ aepting a slight approximation (Biano 00 i.e.: A Φ( i, e = A ( C C + ( C + C d sin( θ + β A A C π A C A C arsin( A γ d + A sin( θ + β γ A γ + ( A γ d ( A γ d + A C d + A C } γ d A are given by Eq. (, ( Ru ( Ru = os( λ Ru sin( λ Ru ( sin( λ os( λ where A, A and C C C C Ru Ru Ru = by Eq. ( and: C = C C + C ( A A A Φ ( = arsin + sin ( θ + β sin ( θ + β sin ( θ + β γ d ( Ru = γ = sin ( θ + β Case o shallow onrete rature and strip s ultimate resisting bond length equal to the eetive bond length (u = ( ( (

16 0 In this ase, the imum eetive apaity is attained or a value o the CDC opening angle γ slightly larger than γ = ( d sin ( θ + β at whih the end slip ours at the end o the CDC (Biano 00. Anyway, sine the expressions o i, e ( γ are very omplex or γ < γ γ, instead o arrying out the derivative ( d i, e ( γ dγ = 0 to searh or the exat value o γ, it is deemed reasonable to assume γ as angle where the imum eetive apaity ours. The solution so obtained, slightly underestimating the real imum, is: A C i, e = A C d γ + d A γ π ( arsin( A γ d ( A γ d ( A γ + d } where A, A and A are given by Eq. ( and: γ = γ = sin ( θ + β ( d Case o shallow onrete rature and strip s ultimate resisting bond length larger than the eetive bond length (u = In this ase, the imum eetive apaity is attained or a value o the CDC opening angle γ slightly larger than γ = ( sin ( θ + β at whih the end slip ( Ru ours at the end o the CDC (Biano 00. Again, sine the expressions o i, e ( γ are very omplex or γ < γ γ, it is deemed a reasonable ompromise between auray o predition and omputational demand, to assume γ as angle in orrespondene o whih the imum eetive apaity ours. The solution so obtained, slightly underestimating the real imum, is: C A Φ ( A C π i, e = A C + + d sin( θ + β A A γ ( + d γ sin( θ + β where A, A and A are given by Eq. (, Φ ( as given by Eq. ( and: γ d ( Ru = γ = sin ( θ + β Case o deep onrete rature (u = In this ase, it is not possible to tell a priori i the imum eetive apaity is attained at a value o the CDC opening angle suh as to yield an imposed end slip at the end o the rak, equal to (Biano 00. Thus, the or to ( Ru imum eetive apaity will be given by: { } i, e i, e ; i, e u = ( where: d (0

17 0 A C i, e = i, e ( γ = A C d γ + d A γ π arsin( A γ d ( A γ d ( A γ + d } u γ = γ = d sin ( θ + β and: = ( γ = i, e i, e u A Φ ( u A C π A ( C C + ( C + C + d sin( θ + β A A A C u A C + arsin( A γ d + sin( θ + β γ A γ + ( A γ d ( A γ d + A C d + A C γ d } γ d ( Ru = γ = sin ( θ + β and where A, A and by Eq. ( and Φ ( as given by Eq. (. u A are given by Eq. (, C ( Ru and ( Ru ( ( ( ( C as given ACTUA AND DESIGN AUE OF THE SHEAR STRENGTHENING CONTRIBUTION The atual and design value o the NSM shear strength ontribution, an be d obtained as ollows: l ( d = N,int i, e sin β γ = γ ( Rd Rd where γ Rd is the partial saety ator, divisor o a apaity, that an be assumed as.-. aording to the level o unertainty aeting the input parameters but, in this respet, a reliability-based alibration is needed. MODE APPRAISA The proposed model was herein applied to the T ross-setion RC beams tested by Dias and Barros (00. The beams tested were RC beams haraterized by the same test set-up with the same ratio between the shear span and the beam eetive depth ( a d =., the same amount o longitudinal reinorement, the same kind o CFRP strips and epoxy adhesive. The details o the beams taken to appraise the preditive perormane o the developed model are listed in Table. Those beams are haraterized by the ollowing ommon geometrial and mehanial parameters: b = 0 mm ; h = 00 mm ; = MPa ; =. MPa ; w E w u =. GPa ; a =. mm ; b =.0 mm ( mm = 0.0 in - N = 0. lb - 00 psi =. MPa. The CDC inlination angle θ adopted in the simulations, m

18 0 0 listed in Table or all the beams analyzed, is the one experimentally observed by inspeting the rak patterns. The angle α was assumed equal to., being the average o values obtained in a previous investigation (Biano et al. 00 by bak analysis o experimental data. The parameter haraterizing the loading proess is: ɺ = rads, whih guarantees a good ompromise between auray o predition and omputational demand. Conrete average tensile strength tm was alulated rom the average ompressive strength by means o the ormulae o the CEB Fib Model Code resulting in. MPa. The parameters haraterizing the adopted loal bond stress-slip relationship (Fig. b are: τ 0 = 0. MPa and =. mm (Biano 00. Those values were obtained by the values haraterizing the more sophistiated loal bond stress-slip relationship adopted in previous works (Biano et al. 00a, 0, by ixing the value o τ 0 = 0. MPa and determining =. mm by equating the rature energy. In this respet, it has to be underlined that the neessity is elt to develop rigorous equations that would allow the values ( τ 0, haraterizing the loal bond stress slip relationship to be determined on the basis o: a superiial hemial and miro-mehanial properties o FRP, adhesive and onrete, and b the adhesive layer thikness. Nonetheless, urther researh is, in this respet, required. Table shows that the model, in general, provides reasonable underestimates o the exp exp experimental reordings sine the ratio presents mean value and standard deviation equal to 0. and 0., respetively. The values o NSM shear strength ontribution have also been ompared with the imum values provided by the more reined model in orrespondene o three dierent geometrial onigurations that the ourred CDC ould assume with respet to the strip (,, and,, in Table. The simpliied model herein presented, in some ases (e.g. beam S- provides a value o the NSM shear strength ontribution that lies in between the minimum and imum values obtained by the more reined model and in other ases (e.g. S-I that is rather lower than the lower bound o the values obtained by the more reined model. This is reasonable, sine the approximations introdued inevitably redue the auray. Table alues o the parameters haraterizing the beams adopted to appraise the ormulation proposed ( mm = 0.0 in - N = 0. lb - 00 psi =. MPa. Beam abel exp θ β s mm Steel Stirrups, kn, kn, kn exp kn mm u kn S- 0 0 F/ S S S-I S-I S-I *.. S-I S-I S-I

19 The model herein proposed, as the more reined one, both seem to provide reasonable estimates o the experimental reordings regardless o the amount o existing stirrups. Atually, the authors think that the amount o existing stirrups aets the depth to whih the onrete rature an penetrate the beam web ore but, sine it exp also aets the CDC inlination angle θ, both models end up giving satiatory results regardless o the amount o existing stirrups (Table. Anyway, in this respet, urther researh is needed. CONCUSIONS A losed-orm design proedure to evaluate the NSM FRP strips shear strength ontribution to RC beams was developed by simpliying a more sophistiated model reently developed. That proedure was obtained by introduing some substantial simpliiations, suh as: a assuming a simpliied loal bond stress-slip relationship, b taking into onsideration the average-available-bond-length NSM FRP strip onined to a onrete prism, and assuming the onrete rature suraes as being semi-pyramidal instead o semi-onial. Given those simpliiations, the proedure is based on the evaluation o the onstitutive law o the average-available-bond-length strip and the determination o the imum eetive apaity that this latter an provide during the loading proess o the strengthened beam, one the kinemati mehanism has been suitably imposed. The most ompliate task is the orret evaluation o the single average-available-bond-length strip s omprehensive onstitutive law, but it an be easily arried out by means o the inormatis tools available to every strutural engineer nowadays. The estimates o the NSM shear strength ontribution obtained by means o that simpliied model showed a reasonable agreement with both the experimental reordings and the preditions obtained by a more sophistiated model. Anyway, the introdution o substantial simpliiations inevitably brought a loss o auray. Moreover, many aspets suh as the orret evaluation o the loal bond stress slip relationship and the issue o the interation with existing stirrups still have to be addressed. ACKNOWEDGEMENTS The authors o the present work wish to aknowledge the support provided by the Empreiteiros Casais, S&P, degussa Portugal, and Seil (Unibetão, Braga. The study reported in this paper orms a part o the researh program CUTINEMO - Carbon iber laminates applied aording to the near surae mounted tehnique to inrease the lexural resistane to negative moments o ontinuous reinored onrete strutures supported by FCT, PTDC/ECM/0/00. Also, this work was arried out under the auspies o the Italian DPC-Reuis Projet (repertory n. 0, Researh ne, whose inanial support is greatly appreiated. REFERENCES Biano,., Barros, J.A.O., Monti, G., (00. Shear Strengthening o RC beams by means o NSM laminates: experimental evidene and preditive models, Tehnial report 0-DEC/E-, Dep. Civil Eng., Shool Eng. University o Minho, Guimarães- Portugal. Biano,., Barros, J.A.O., Monti, G., (00. Shear Strengthening o RC beams by means o NSM strips: a proposal or modeling debonding, Tehnial report 0- DEC/E-, Dep. Civil Eng., Shool Eng. University o Minho, Guimarães- Portugal.

20 Biano,., (00. Shear Strengthening o RC beams by means o NSM FRP strips: experimental evidene and analytial modeling, PhD Thesis, Dept. o Strutural Engrg. and Geotehnis, Sapienza University o Rome, Italy, submitted on Deember 00. Biano,., Barros, J.A.O., Monti, G., (00a. Three dimensional mehanial model or simulating the NSM FRP strips shear strength ontribution to RC beams, Engineering Strutures, (, April 00, -. Biano,., Barros, J.A.O., Monti, G., (00b. Bond Model o NSM FRP strips in the ontext o the Shear Strengthening o RC beams, ASCE Journal o Strutural Engineering, (, June 00. Biano,., Barros, J.A.O., Monti, G., (0. New approah or modeling the ontribution o NSM FRP strips or shear strengthening o RC beams, ASCE Journal o Composites or Constrution, (, January/February 0. CEB-FIP Model Code 0, ( Bulletin d Inormation N /, Final version printed by Th. Telord, ondon, (; ISBN 0---; 0 pages. Dias, S.J.E. and Barros, J.A.O., (00. Shear Strengthening o T Cross Setion Reinored Conrete Beams by Near Surae Mounted Tehnique, Journal o Composites or Constrution, ASCE, ol., No., pp Monti, G., Renzelli, M., uiani, P., (00 FRP Adhesion to Unraked and Craked Conrete Zones, Proeedings o the th International Symposium on Fibre- Reinored Polymer (FRP Reinorement or Conrete Strutures (FRPRCS-, Singapore, July, -. Monti, G., Santinelli, F., otta, M.A., (00 Mehanis o FRP Shear Strengthening o RC beams, Pro. ECCM, Rhodes, Greee. Monti, G., otta, M.A., (00 Tests and design equations or FRP-strengthening in shear, Constrution and Building Materials (00, (, April 00, -0. Mohammed Ali, M.S., Oehlers, D.J., Seraino, R. (00. ertial shear interation model between external FRP transverse plates and internal stirrups, Engineering Strutures, -. Mohammed Ali, M.S., Oehlers, D.J., Griith, M.C., Seraino, R. (00. Interaial stress traner o near surae-mounted FRP-to-onrete joints, Engineering Strutures 0, -. Rizzo, A. and De orenzis,., (00 Behaviour and apaity o R beams strengthened in shear with NSM FRP reinorement, Constrution and Building Materials, ol., n., April 00, -. Yuan, H., Teng, J.G., Seraino, R., Wu, Z.S., Yao, J. (00. Full-range behavior o FRP-to-onrete bonded joints, Engineering Strutures,, -. 0