THEORETICAL MODEL AND COMPUTATIONAL PROCEDURE TO EVALUATE THE NSM FRP STRIPS SHEAR STRENGTH CONTRIBUTION TO A RC BEAM

Transcription

1 THEORETICA MODE AND COMPUTATIONA PROCEDURE TO EAUATE THE NSM FRP STRIPS SHEAR STRENGTH CONTRIBUTION TO A RC BEAM inenzo Biano 1, Giorgio Monti and J.A.O. Barros 3 Abstrat: 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 the 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: FRP; NSM; Computational Proedure; Shear Strengthening; Conrete Frature; Debonding; Tensile Rupture. Introdution Shear strengthening o RC beams by NSM tehnique onsists o gluing 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 008, Biano et al. 009a-b and 010. 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 1 Post Dotoral Student, Dept. o Strutural Engrg. and Geotehnis, Sapienza University o Rome, via A. Gramsi 53, Rome, Italy. vinenzo.biano@uniroma1.it, orresponding Author. Tel , Fax Full Proessor, Dept. o Strutural Engrg. and Geotehnis, Sapienza University o Rome, via A. Gramsi 53, Rome, Italy. giorgio.monti@uniroma1.it. 3 Assoiate Proessor, Dept. o Civil Engineering, University o Minho, Campus de Azurém, Guimarães, Portugal. barros@ivil.uminho.pt. 1

2 proedure that has to be: a mehanially-based and b simple to implement. As to the rst point, it has to ulll equilibrium, kinemati ompatibility and onstitutive laws. As to the seond point, it has to be a design tool that is easy to apply. For this purpose, a reasonable ompromise between auray o predition and omputational demand has to be ahieved. Exessively simplied 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 simpliations (Biano 008: 1 a bi-linear rigid-sotening loal bond stress-slip diagram is adopted instead o a multilinear diagram, onrete rature surae is assumed as semi-pyramidal instead o semionial, 3 attention is oused on the average-available-bond-length NSM FRP strip glued on the relevant prism o surrounding onrete, 4 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. (003, and 5 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. 004, Monti and otta 007. The main eatures o the resulting modeling strategy are reported hereater. During the loading proess o a RC beam subjet to shear, when onrete average tensile strength tm is attained at the web intrados (Fig. 1, 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. 1a. The CDC an be shematized as an inlined plane dividing the web into two portions sewn together by the rossing strips (Fig. 1a. At load step t 1, 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. 1a. From that step on, by inreasing the applied load, the CDC opening angle γ ( t n progressively widens (Fig. 1a. The strips rossing the CDC oppose its widening by anhoring to the surrounding onrete to whih they traner, by bond, the ore originating at their intersetion with the CDC, l O i, as a result o the imposed end slip [ γ ( t ]. The apaity o eah strip is provided by its available bond length that is the n shorter between the two parts into whih the rak divides its atual length (Fig. 1a. Bond is the mehanism through whih stresses are tranerred to the surrounding onrete (Yuan et al. 004, Mohammed Ali et al. 006 and 007, Biano et al The loal bond stress-slip relationship τ (, omprehensively simulating the mehanial phenomena ourring at 1 the strip-adhesive interae, within the adhesive layer and at 3 the

3 adhesive-onrete interae, an be represented, in a simplied way, by a bi-linear urve (Fig. 1b. 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. 1b (Biano 008. The rst rigid branh (0-τ 0 represents the overall initial shear strength o the joint, independent o the deormability o the adhesive layer and attributable to the miro-mehanial and mainly hemial properties o the involved materials and relative interaes. In at, the parameter τ 0 is the average o the ollowing physial entities enountered in sequene by stresses lowing rom the strip to the surrounding onrete, i.e.: adhesion at the strip-adhesive interae, ohesion within the adhesive itsel, and adhesion at the adhesive-onrete interae (e.g. Sekuli and Curnier 006, Zhai et al The τ ( urve adopted (Fig. 1b envisages that, by imposing inreasing end slips to the FRP strip, raks orm instantaneously within the adhesive layer, both orthogonally to the (inlined tension isostatis and along the strip-adhesive and adhesive-onrete interaes (e.g. Sena-Cruz and Barros 004. Stresses are tranerred by rition and miro-mehanial interlok along those miro-raks. Nonetheless, by imposing inreasing end slips, those raks progressively beome smoother (sotening rition phase up to the point ( = 1 in whih rition an no longer be mobilized and the strip is pulled out without having to overome any restraint let (ree slipping phase. The onstitutive law ( R ; o an NSM FRP strip, i.e. the ore transmissible by a strip with resisting bond length R 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. 1a,-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 (Dias and Barros 008, Rizzo and De orenzis 009 is taken into aount in a simplied 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, sine the larger the amount o stirrups, the shallower onrete rature is expeted to be (Biano et. al 006 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. 1-d. 3

4 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. 1e. 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 adhesive-onrete interaes, so that the strip pulling out results (Fig. 1e. 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. 1-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 redue the resisting bond length R that is the portion o the initial available bond length 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. 1e. Moreover, regardless o an initial onrete rature, the strip an rupture (Fig. 1e. The ormulation obtained by this strategy is presented in the ollowing setions along with the main mehanial bases. Calulation proedure The input parameters inlude (Fig. : beam ross-setion web s depth h w and width w b ; inlination angle o both CDC and strips with respet to the beam longitudinal axis, θ and β, respetively; strips spaing measured along the beam axis s ; angle α between axis and prinipal generatries o the semi-pyramidal rature surae (Fig. 1-d; onrete average ompressive strength m ; strips tensile strength u and Young s modulus E ; thikness a and width b o the strip ross-setion; inrement stress τ 0 and slip 1 dening the adopted loal bond stress-slip relationship (Fig. 1b: ɺ o the imposed end slip; values o bond τ 1 0 τ ( < = 1 0 > The geometrial onguration is adopted in whih the minimum integer number l,int (1 N o strips ross the CDC with the rst one plaed at a distane equal to s rom the rak origin (Fig. 1a. This onguration orresponds 4

5 to the minimum o the sum o all the available bond lengths number to the lowest integer, as ollows:. l,int N is obtained by rounding o the real N l,int = round o hw ( otθ + ot β s ( and the average available bond length is obtained by: 1 = (3 N l,int l N,int i= 1 with: sinθ hw i s or x < (ot θ + ot β = sin( θ + β sinθ hw i s or x (ot θ + ot β sin( θ + β (4 and: x = i s (5 Ater having dened the geometrial harateristis o the simple strutural system omposed o the averageavailable-bond-length strip within the relevant prism o surrounding onrete, it is neessary to determine its onstitutive law ( ; One and the orresponding imum eetive apaity, as explained hereater. has been obtained, the atual and design d values o the NSM shear strength ontribution an be obtained by Eq. (36. Constitutive law o a single NSM FRP strip The mehanial behaviour o an FRP strip glued near the surae o a onrete prism and subjeted to an inreasing imposed end slip is very omplex. That omplexity is due to the interation between the mehanism o ore traner to the surrounding onrete through bond stresses, mobilized along the glued surae, and the possibility o onrete rature. Due to this interation, 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 008. In at, or eah t n, that system undergoes modiations up to reahing the equilibrium onguration q e. Whenever onrete ratures, the mehanism o ore traner to the 5

6 surrounding onrete leaps orward towards the strip s ree end. In general, in orrespondene o eah leap, the overall traner length tr, ( R; inreases and the resisting bond length dereases (Fig. 1-d. Thus, in general, at eah leap, onrete tensile rature apaity inreases and at the same time the bond-tranerred ore dereases, until equilibrium is attained. In this senario, in order to determine the omprehensive onstitutive law ( ; 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 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 modied aordingly. n Bond-based onstitutive law The bond behaviour o an NSM FRP strip subjet to an inreasing imposed end slip an be modelled by ullling equilibrium, kinemati ompatibility and onstitutive laws o both adhered materials (onrete and FRP and loal bond between themselves (Biano 008. In this way, it is possible to obtain losed-orm analytial equations or both the bond-based onstitutive law ( R ; traner length tr, ( R; o a single strip and the orresponding bond. The latter two quantities represent: the ore a strip o resisting bond length R an traner by bond, as untion o, and the orresponding amount o R along whih bond is mobilized, respetively. One the invariant distribution o shear stress τ ( x and slip ( x is determined or an innite value o (Fig. 3a,, and R tr an be determined or any nite value o R (Fig. 3b-g onsidering the migration o τ ( x along R, rom the loaded end (E to the ree end (FE, by inreasing equations o tr, ( R ; and ( R ;. The analytial, presented below and plotted in Fig. 4, an be determined by: a onsidering the relative position o with respet to the invariant distribution o τ ( x and ( x, and b R integrating τ ( x, respetively (Biano 008, Biano et. al 009b. Those analytial equations envisage (Figs. 3-4, or a given R, three phases, whose limits ( 1; ; 3 are untion o the value assumed by R with respet to the eetive bond length tr1 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. In the rst phase (Figs. 4,3b, the ore transmitted by bond stresses inreases up to reahing the peak in orrespondene o 6

7 that is (Fig. 3, or R tr1 1 ( R <, the or whih the invariant distribution o bond stresses has reahed the strip s ree end while, or R tr1, the or whih tr, = tr1. The seond phase (Figs. 4,3d, or R < tr1, is haraterized by a derease o and a onstany o tr, while, or R tr1, remains onstant and equal to the peak 1 and, tr goes on inreasing up to ( R. ( R is (Figs. 4,3e, or R < tr1, the or whih the null value o the invariant distribution o τ ( x has reahed the loaded end while, or R tr1, the or whih the value τ 0 o the invariant distribution o τ ( x has reahed the ree end. In the third phase (Figs. 4,3, the invariant distribution o bond stresses progressively moves away rom o tr, that is equal to R R, regardless o its value, resulting in a derease o up to zero and in a onstant value is (Figs. 4,3g, or eah R tr1, the in orrespondene o whih. 3 ( R the null value o the invariant distribution o τ ( x has reahed the ree end. Note that, or ontinuity, or = it is ( = ( and the seond phase redues to a point (Fig. 4. R tr1 1 tr1 tr1 1 The bond traner length is as ollows: 1 λ tr, ( R; = tr ( = aros 1 λ τ 0 J1 ( ; ( ; = + s ( tr, R tr1 R < = tr, R tr1 tr1 tr 1( 0.0 R ( < ( 1 R R ( = ( R < 3 ( R tr, R; R (6 tr, ( R ; = 0.0 > 3 ( R and the bond-based onstitutive law: ( ; = 3 λ { 1 os( λ ( 1 sin( λ ( } ( R R p tr tr J C C R tr p J C x C x R tr = ( ( < tr 1; = 3 λ 1 os( λ sin( λ ( ( 1; 1 tr R ( < ( 1 R R (7 tr 1 ( ; = 3 λ 1 os( λ sin( λ s ( ( + ( R < 3 R R p J C x C x tr1 tr R ( R ; = 0.0 > 3 ( R where: 7

8 = b + a (8 p is the eetive perimeter o the strip ross-setion, and: A E E A = ; J = + ; = λ τ 0 J1 A E A E E A + E A E A E A τ J τ J J C C p 1 J = ; 1 = 1 ; = p ( A E + A E λ λ (9 are bond-modeling onstants (Biano 008, Biano et al. 009b, with Ap = a b and A = s bw the ross-setion o the strip and the onrete prism, respetively. Moreover, the eetive bond length tr1 and the orresponding imum bond ore 1 are given by: π τ 0 J1 tr1 = ; 1 = p J3 λ 1 λ λ (10 The value o resisting bond length undergoing sotening rition, as untion o the imposed end slip, is given by: tr 1 λ ( = aros 1 λ τ 0 J1 (11 and the value o resisting bond length undergoing ree slipping: A J ( 1 ( = (1 s tr 1 The resisting bond length-dependent values o imposed end slip dening the extremities o the three bond phases, are given by (Fig. 3b-g: 1 ( R ( τ 0 J1 C ( λ C ( λ = λ or 1 sin R + os R + or R < tr1 1 R tr1 or < = + 1 R tr1 R 1 1 ( R tr1 or R tr1 A J ( R 3 1 = + A R J 1 (13 (14 (15 Conrete tensile rature apaity 8

9 The onrete tensile rature apaity ( tr, is obtained by spreading the onrete average tensile strength over the semi-pyramidal surae (Fig. 1-d o height equal to the total traner length, tm tr, orthogonally to it in eah point. By integrating one obtains: bw ( tr, = tm min{ tr, tan α; } sin ( θ + β s sin β tr, sinα s sin β tr, sinα min ; + min ; sin ( θ + β sin ( θ + β + α sin ( θ + β sin ( θ + β α (16 where tm an be determined rom the average ompressive strength. The total traner length is evaluated as reported in next Eq. (17. Comprehensive onstitutive law At the t n load step, an iterative proedure ( qm : q1 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 ( t n and resisting bond length R ( tn; q m (Figs In partiular, at the q m iteration o the t n load step, based on R ( tn; q m and ( t n, the bond traner length tr, R ( tn; qm ; ( tn and the orresponding bond-tranerred ore R ( tn; qm ; ( tn are evaluated as reported in Eq. (6 and Eq. (7, respetively. Then, the urrent value o the total traner length is evaluated as ollows: tr, ( tn; qm = ( tn 1 ; qe + tr, R ( tn; qm ; ( tn + ( tn; qm (17 where ( t q is the umulative depth o the onrete rature surae resulting rom the equilibrium o the n 1 ; e preeding tn 1 load step and ( tn; qm is the inrement o onrete rature depth orresponding to the urrent t n, aumulated up to the urrent m q (Fig. 6: q m ( 1 tn; qm = tr, R ( tn; qm ; ( tn (18 q 1 Then, ater having evaluated the onrete rature apaity ( tr, as indiated in Eq. (16, i it is: R ( tn; qm ; ( tn tr, ( tn; qm (19 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 ( t ; q + and R n m 1 9

10 ( tn; q m + 1 are updated ( R ( tn; qm+ 1 = R ( tn; qm tr, R ( tn; qm ; ( tn, ( tn; qm+ 1 = ( tn; qm + tr, R ( tn; qm ; ( tn and iteration is perormed ( q m + 1 (Fig. 5. At eah o those leaps, the point representative o the strip state moves rom one bond-based onstitutive law R ( tn; qm ; to the other R ( tn; qm+ 1 ; and, as long as the updated value o R is larger or equal to the neessary bond traner length [ ( t ], suh leap is only visible in a three dimensional tr n representation (Fig. 6d. The neessary bond traner length [ ( t ] is the bond traner length that would tr n be neessary, i R were innite, to transmit the orresponding ore to the surrounding onrete, with tr [ ( tn ] = tr [ ( tn ] or ( t n 1 and [ ( ] s tr tn = tr1 + tr [ ( tn ] or ( t n > 1 (Fig. 3. Note also that, at eah q iteration, the equality ( t ; q + ( t ; q + ( t ; q = ( t ; q = has to be m R n m n 1 e n m R 1 1 ullled (Fig. 6 and Fig. 7. More in detail, at the q m iteration o the n t load step, i onrete is not in equilibrium ( = 0, one o the e ollowing alternatives might our: onrete rature is deep ( d = 1 but it does not reah the ree end, i.e. the updated resisting bond length ( t ; q + is not long enough to mobilize, or the urrent ( t, a bond traner length as R n m 1 n large as the neessary one: < [ ( t ] (Fig. 5. Note that in this ase, the passage o the point R tr n representative o the strip state rom one bond-based onstitutive law to the other is also visible in a bi-dimensional representation (Fig. 7; onrete rature is deep ( d = 1 and it reahes the ree end, i.e. the updated resisting bond length ( t ; q + is null. Note is taken o the urrent value o the imposed end slip ( ( t and the R n m 1 u n inremental proedure is terminated sine a deision about the omprehensive onstitutive law an already be taken ( u = 1 (Fig. 8a. On the ontrary, i at the q m iteration o the t n load step, onrete is in equilibrium ( e = 1, 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 tensile rupture apaity ( tr. The inremental proedure is terminated sine, even i the surrounding onrete is in 10

11 equilibrium, the strip has ruptured ( u = and note is taken o the ultimate imposed end slip ( ( t ; u n n+ 1 the next value o the imposed end slip ( t 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+ 1 1 R ( tn; qe. Sine starts to derease or ( t n+ 1 (Figs. 6-7, the inremental proedure is terminated and note is taken o the urrent value o the resisting bond length ( Ru R and o its relationship with the eetive bond length tr1 ( u 3 i Ru < tr1, u 4 i Ru = tr1 or u 5 i Ru > tr1 ; onrete rature is deep ( d = 1 and it does not reah the ree extremity. The inremental proedure is terminated ( u = 6 (Fig. 8d; the next value o the imposed end slip ( t + is smaller than the one where the peak bond ore is n 1 attained or the urrent value o the resisting bond length ( tn+ 1 < 1 R ( tn; qe. Then, the imposed end slip is inremented and the iteration arried out. The inremental proedure desribed above is terminated and, depending on the phenomenon haraterizing the spei ase at hand and the type o onstitutive law assoiated ( u, the parameters neessary to dene ( ; are returned, i.e.: deep onrete rature that reahes the strip s ree extremity ( u = 1 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 ( ; ours. ( ; o Eq. (7 or 0.0 u (Fig. 8a; is given by the rst bond phase shallow or absent onrete rature with an ultimate value o resisting bond length smaller ( u = 3, equal ( u = 4 or larger ( u = 5 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. ( ; is given by Eq. (7 or R Ru = (Fig. 8b-; deep tensile rature with an ultimate value o resisting bond length very short but not null ( u = 6. The parameters neessary to determine the omprehensive onstitutive law are both the imposed end slip 11

12 in orrespondene o whih the peak o ( ; u ours and the ultimate value assumed by the resisting bond length Ru. ( ; is given by: the rst bond phase o Eq. (7 or 0.0 u, the seond bond phase o Eq. (7 or u ( Ru ( ( < (Fig. 8d. Ru 3 Ru < and the third bond phase o Eq. (7 or Maximum eetive apaity o a single NSM FRP strip The eetive apaity ( γ is the average o the NSM FRP strip apaity along the CDC ( γ ξ or, e, CDC ; a given value o the CDC opening angle γ (Figs. 9-13, where ξ is the reerene system assumed along the CDC (Fig. 1a. ( γ ξ is obtained by introduing the kinemati ompatibility, CDC ; ( ( 1 γ ; ξ = ξ γ sin ( θ + β, with ξ [ 0 h w sin( θ ] into the omprehensive onstitutive law o the single average-available-bond-length NSM FRP strip ( ;. For the sake o brevity, all o the details are herein omitted but they an be ound elsewhere (Biano 008. The equation to evaluate the imum eetive apaity, 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 spei ase at hand. Cases o onrete rature that reahes the strip s ree extremity (u = 1 or strip tensile rupture (u = 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. 9 i.e.: 1 A C, e =, e ( γ = A1 C1 d γ + d A3 γ π arsin( 1 A3 γ d ( 1 A3 γ d 1 ( 1 A3 γ + d } (0 where: 3 p J3 λ sin( θ + β A1 = 4 τ J 0 1 ; p 3 A = J λ ; λ sin( θ + β A3 = τ J 0 1 (1 are integration onstants independent o the type (u o omprehensive onstitutive law and: γ = γ = 1 d u sin ( θ + β ( 1

13 Case o shallow onrete rature and strip ultimate resisting bond length smaller than the eetive bond length (u = 3 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 sake o simpliity, it is assumed that (Fig. 10 i.e.:. For the is eetively attained or γ aepting a slight approximation 1 1 A Φ1 ( 1 A C π A C1 1 1, e = A1 ( C1 C1 + ( C + C + d sin( θ + β A3 4 A3 sin( θ + β γ (3 A C arsin( 1 A3 γ d ( 1 A3 γ d 1 ( 1 A3 γ d + + A C1 d + A1 C A γ 1 γ d 3 where A 1, A and 3 A are given by Eq. (1, 1 1 ( Ru = by Eq. (13 and: ( Ru = os( λ Ru sin( λ Ru ( = sin( λ + os( λ C C C C C C C C Ru 1 Ru Ru (4 A3 A3 A3 Φ 1 ( = arsin sin ( θ + β sin ( θ + β sin ( θ + β γ d ( Ru = γ = sin ( θ + β (5 (6 Case o shallow onrete rature and strip s ultimate resisting bond length equal to the eetive bond length (u = 4 In this ase, the imum eetive apaity is attained or a value o the CDC opening angle γ slightly larger than γ1 = 1 ( sin ( θ + β at whih the 1 end slip ours at the end o the CDC (Fig. 11. Anyway, sine d the expressions o, e ( γ are very omplex or γ1 < γ γ, instead o arrying out the derivative ( d, e ( γ dγ = 0 to searh or the exat value o γ, it is deemed reasonable to assume γ 1 as the angle where the imum eetive apaity ours. The solution so obtained, slightly underestimating the real imum, is: 13

14 1 A C, e = A1 C1 d γ + d A3 γ π arsin( 1 A3 γ d ( 1 A3 γ d 1 ( 1 A3 γ + d } (7 where A 1, A and A 3 are given by Eq. (1 and: γ = γ = 1 d 1 sin ( θ + β (8 Case o shallow onrete rature and strip s ultimate resisting bond length larger than the eetive bond length (u = 5 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 d ours at the end o the CDC (Fig. 1. Again, sine the expressions o, e ( γ are very omplex or γ < γ γ 3, it is deemed a reasonable ompromise between auray o predition and omputational demand, to assume γ as the angle in orrespondene o whih the imum eetive apaity ours. The solution so obtained, slightly underestimating the real imum, is: 1 1 C A Φ1 ( 1 A C π 1 1, e = A1 C d d sin( θ + β A3 4 A3 γ γ sin( θ + β (9 where A 1, A and A 3 are given by Eq. (1, Φ 1 ( 1 as given by Eq. (5 and: γ d ( Ru = γ = sin ( θ + β (30 Case o deep onrete rature (u = 6 In this ase, it is not known 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 the imum eetive apaity will be given by: { } 1, e, e ;, e or to ( Ru u (Fig. 13. Thus, = (31 where: 14

15 1 1 A C, e =, e ( γ 1 = A1 C1 d γ 1 + d A3 γ 1 π arsin( 1 A3 γ 1 d ( 1 A3 γ 1 d 1 ( 1 A3 γ 1 + d } γ = γ = 1 1 d u sin ( θ + β (3 (33 and: = ( γ =, e, e 1 u A Φ1 ( u A C π A C1 u 1 A1 ( C1 C1 + ( C + C + d sin( θ + β A3 4 A3 sin( θ + β γ A C arsin( 1 A3 γ d + ( 1 A3 γ d 1 ( 1 A3 γ d + A C1 d + A1 C1 γ d A γ 3 (34 γ d ( Ru = γ = sin ( θ + β (35 and where A 1, A and 3 given by Eq. (5. A are given by Eq. (1, C1 ( Ru and ( Ru C as given by Eq. (4 and Φ ( as 1 u Atual and Design value o the Shear Strengthening Contribution The atual and design value d o the NSM shear strength ontribution, an be obtained as ollows: 1 1 l ( d = N,int, e sin β γ = γ (36 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. Model Appraisal The proposed model was applied to the RC beams tested by Dias and Barros (008, by Dias et al. (007 and by Dias (008. The beams tested in the rst two experimental programs (series I and II were T ross-setion RC beams haraterized by the same test set-up with the same ratio between the shear span and the beam eetive depth ( a d =.5, the same amount o longitudinal reinorement, the same kind o CFRP strips and epoxy adhesive and they diered or the onrete mehanial properties. In at, the rst experimental program (series I was haraterized by a onrete average ompressive strength m o 31.1 MPa, while the seond (series II by 18.6 MPa. Both series presented dierent ongurations o NSM strips, in terms o both inlination β and 15

16 spaing s. The rst program also inluded beams haraterized by a dierent amount o existing steel stirrups (see Table 1. The beams tested in the third experimental program (series III were haraterized by the same test set up, but with a dierent shear aspet ratio ( a d = 3.3 and distint onrete mehanial properties ( = 59.4 MPa. Some o them were also subjeted to pre-raking (their label inludes a letter F. The details m o the beams taken to appraise the preditive perormane o the developed model are listed in Table 1. Those beams are haraterized by the ollowing ommon geometrial and mehanial parameters: b = 180 mm ; hw = 300 mm ; u = 95 MPa (or the series I and II and u = 848 MPa (or the series III; w E = GPa (or the series I and II and E = GPa (or the series III; a = 1.4 mm ; b = 10.0 mm. The CDC inlination angle θ adopted in the simulations, listed in Table 1 or all the beams analyzed, is the one experimentally observed by inspeting the rak patterns (Dias 008. Note that the experimental observations onrm the expeted trend aording to whih θ diminishes or inreasing values o the ratio a d (e.g. Bousselham and Chaalal 004, Chao et. al In at, or some beams o the III series ( a d = 3.3, assumes values smaller than 45º and up to 0º (Table 1. In this respet, it has to be stressed that assuming θ = 45º an result exessively onservative sine, with respet to smaller values (e.g. θ = 0º, and other parameters being the same, the predited NSM shear strength ontribution dereases due to the at that the number o strips eetively rossing the CDC diminishes (Biano 008. It would be neessary to develope rigorous equations to evaluate the CDC inlination angle θ as untion o 1 shear aspet ratio a d and amount o both NSM strips and 3 existing steel stirrups but, in this respet, urther researh is neessary. The angle α was assumed equal to 8.5, being the average o values obtained in a previous investigation (Biano et al. 006 by bak-analysis o experimental data. As to the value o α, due to its importane to the predition auray o NSM shear strength ontributions, urther researh is desirable. The parameter haraterizing the loading proess is: ɺ = rads. Conrete average tensile strength tm was alulated rom the average ompressive strength by means o the ormulae o the CEB Fib Model Code 1990 resulting in.45 MPa, 1.45 MPa and 4.17 MPa or the series I, II and III, respetively. The parameters haraterizing the adopted loal bond stress-slip relationship (Fig. 1b are: τ 0 = 0.1 MPa and 1 = 7.1 mm. Those values were obtained by the values haraterizing the more sophistiated loal bond stress-slip relationship adopted in previous works (Biano et al. 009a, 010, by xing the value o τ 0 = 0.1 MPa and determining 1 = 7.1 mm by equating the rature energy. In this respet, it has to be underlined that the neessity is elt to exp θ 16

17 develop rigorous equations that would allow the values ( τ, haraterizing the loal bond stress slip relationship to be determined on the basis o: a superial hemial and miro-mehanial properties o FRP, adhesive and onrete, and b the adhesive layer thikness. Nonetheless, urther researh is, in this respet, required. However, as highlighted by means o parametri studies (Biano 008, or the values o onrete mehanial properties that an be met in pratie, debonding rarely ours due the high apaity o urrently available strutural adhesives. Thus, slight variations o the values o the parameters τ 0 and 1 an not be elt, in terms o NSM shear strength ontribution, due to the premature ourrene o other ailure modes suh as either onrete rature or strip rupture. For this reason, adopting values o τ 0 = 0.1 MPa and 1 = 7.1 mm or ases haraterized by dierent values o both 1 superial hemial-mehanial properties o FRP, adhesive and onrete, and adhesive thikness, is not expeted to signiantly aet the preditive perormane o the model. Table 1 shows that the model, in general, provides reasonable underestimates o the experimental reordings exp sine the ratio exp 0 1 presents mean value and standard deviation equal to 0.86 and 0.33, respetively. The values o NSM shear strength ontribution have also been ompared with the imum values provided by the more rened model in orrespondene o three dierent geometrial ongurations that the ourred CDC,1 ould assume with respet to the strip (,, and,3 in Table 1. The simplied model herein presented, in some ases (e.g. beam S-5-I provides a value o the NSM shear strength ontribution that lies in between the minimum and imum values obtained by the more rened model and in other ases (e.g. S-5I45-I it gives a value that is rather lower than the lower bound o the values obtained by the more rened model. This is reasonable, sine the approximations introdued inevitably redue the auray. The model herein proposed, as the more rened 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 also aets the CDC inlination angle exp θ, both models end up giving satiatory results regardless o the amount o existing stirrups (Table 1. Anyway, in this respet, urther researh is needed. Conlusions A losed-orm omputational 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 17

18 introduing some substantial simpliations, suh as: 1 assuming a simplied loal bond stress-slip relationship, taking into onsideration the average-available-bond-length NSM FRP strip onned to a onrete prism, and 3 assuming the onrete rature suraes as being semi-pyramidal instead o semi-onial. Given those simpliations, the proedure is based on 1 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 estimates o the NSM shear strength ontribution obtained by means o that simplied model showed a reasonable agreement with both the experimental reordings and the preditions obtained by a more sophistiated model. Anyway, the introdution o substantial simpliations inevitably brought a loss o auray. Moreover, many aspets suh as: 1 the orret evaluation o the loal bond stress slip relationship as untion o both the hemial-mehanial properties o FRP, onrete and adhesive and this latter thikness; the orret evaluation o the CDC inlination angle as untion o shear aspet ratio and amount o both FRPs and existing steel stirrups; and 3 the issue o the interation with existing stirrups, still have to be addressed. Aknowledgements 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 ber 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/73099/006. Also, this work was arried out under the auspies o the Italian DPC-Reuis Projet (repertory n. 540, Researh ne 8, whose nanial support is greatly appreiated. 18

19 Notation A = area o the onrete prism ross setion A = area o the strip s ross setion A 1 = integration onstant entering the expressions to evaluate the A = integration onstant entering the expressions to evaluate the A 3 = integration onstant entering the expressions to evaluate the C 1 = rst integration onstant or the sotening rition phase, e, e, e C = seond integration onstant or the sotening rition phase C1 ( R = resisting-bond-length-dependent integration term C ( R = resisting-bond-length-dependent integration term E = onrete Young s modulus E = strips CFRP Young s modulus J 1 = bond modeling onstant J = bond modeling onstant J 3 = bond modeling onstant d = CDC length = Atual length o th strips = i-th strip available bond length = average available bond length ( t q ; = height o the onrete semi-pyramid in orrespondene o the i-th strip n n p = eetive perimeter o the strip ross setion R ( t q ; = i-th strip resisting bond length n n ( = traner length o the i-th strip or the relevant imposed slip tr, R; ( = neessary bond traner length tr 19

20 ( tr, R; = bond based traner length o the i-th strip or the relevant imposed slip tr1 = imum invariant value o traner length that an undergo elasti phase s tr ( = amount o a traner length or an innite bond length undergoing ree slipping tr ( = sotening ritional amount o a traner length or an innite bond length l,int N = minimum integer number o strips that an eetively ross the CDC OXYZ = rak plane reerene system O = reerene axis along the i-th strip available bond length l l i X i = progressive onrete tensile rature apaity along the i-th strip db 1 = value o ore tranerred by bond along the elasti traner length tr1 = imum eetive apaity exp = experimental value o the NSM shear strengthening ontribution = atual value o the NSM shear strengthening ontribution d = design value o the NSM shear strengthening ontribution ( ξ γ = distribution o the NSM strip apaity along the CDC, CDC ; ( ; = omprehensive onstitutive law o the average available bond length ( ; = bond-based onstitutive law o the average available bond length a = onrete prismati speimen thikness a = strip ross setion s thikness b = onrete prismati speimen width b = strip ross setion s width b w = beam ross setion s width m = onrete average ompressive strength tm = onrete average tensile strength u = FRP strip tensile strength 0

21 h w = beam web height i = strips ounter s x s o x o = reerene axis along the amount o the innite strip in ree slipping phase = reerene axis along the amount o the innite strip in sotening rition phase tr tr i xi o = reerene axis along the strip s traner length q e = iteration in orrespondene o whih equilibrium is attained q m = m-th iteration s = Spaing between adjaent strips along the CDC axis t 0 = load step o ormation o the ritial diagonal rak t 1 = load step at whih the ritial diagonal rak starts widening t n = generi n-th load step t s = slab thikness u = parameter dening the omprehensive onstitutive law type x = position o the i-th strip along the global reerene system Φ ( = Imposed-end-slip-dependent expression 1 α = angle dening the onrete rature surae β = FRP strips inlination angle with respet to the beam longitudinal axis ( x = slip along the strip s length 1 = slip orresponding to the end o sotening rition = imposed slip at the loaded extremity o the i-th strip ɺ = imposed slip inrement = value o dening the end o the rst phase o the bond-based onstitutive law 1 ( R ( R = value o ( R 3 = value o γ ( t n = ritial diagonal rak opening angle dening the end o the seond phase o the bond-based onstitutive law dening the end o the third phase o the bond-based onstitutive law γ 1 = CDC opening angle suh that the imposed end slip at d is equal to 1 1

22 γ = CDC opening angle suh that the imposed end slip at d is equal to γ 3 = CDC opening angle suh that the imposed end slip at d is equal to 3 γ = CDC opening angle or whih the imum eetive apaity is attained γ Rd = partial saety ator divisor o the apaity λ = onstant entering the governing dierential equation or elasti phase θ = ritial Diagonal Crak (CDC inlination angle exp θ = experimentally observed CDC inlination angle τ ( = loal bond stress-slip relationship τ ( x = bond stress along the strip length τ 0 = adhesive-ohesive initial bond strength ξ = reerene axis along the CDC

23 Reerenes Biano,., Barros, J.A.O., Monti, G., (006. Shear Strengthening o RC beams by means o NSM laminates: experimental evidene and preditive models, Tehnial report 06-DEC/E-18, Dep. Civil Eng., Shool Eng. University o Minho, Guimarães- Portugal. Biano,., Barros, J.A.O., Monti, G., (007. Shear Strengthening o RC beams by means o NSM strips: a proposal or modeling debonding, Tehnial report 07-DEC/E-9, Dep. Civil Eng., Shool Eng. University o Minho, Guimarães- Portugal. Biano,., Barros, J.A.O., Monti, G., (009a. Three dimensional mehanial model or simulating the NSM FRP strips shear strength ontribution to RC beams, Engineering Strutures, 31(4, April 009, Biano,., Barros, J.A.O., Monti, G., (009b. Bond Model o NSM FRP strips in the ontext o the Shear Strengthening o RC beams, ASCE Journal o Strutural Engineering, 135(6, June 009. Biano,., Barros, J.A.O., Monti, G., (010. New approah or modeling the ontribution o NSM FRP strips or shear strengthening o RC beams, ASCE Journal o Composites or Constrution, 14(1, January/February 010. Biano,., (008. 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 008. Bousselham A., Chaalal O., (004 Shear Strengthening Reinored Conrete Beams with Fiber-Reinored Polymer: Assessment o Inluening Parameters and Required Researh, ACI Strutural Journal, ol.101, Nº, Marh-April, pp CEB-FIP Model Code 90, (1993 Bulletin d Inormation N 13/14, Final version printed by Th. Telord, ondon, (1993; ISBN ; 460 pages. Chao S.Y., Chen J.F., Teng J.G., Hao Z., Chen J., (005 Debonding in Reinored Conrete Beams Shear Strengthened with Complete Fiber Reinored Polymer Wraps, Journal o Composites or Constrution, ASCE September/Otober 005/1. Dias, S.J.E. (008. Experimental and anlytial researh in the shear strengthening o reinored onreet beams using the near surae mounted tehnique with CFRP strips, PhD Thesis, Department o Civil Engineering, University o Minho, Guimarães-Portugal, in Portuguese. Dias, S.J.E., Biano,., Barros, J.A.O., Monti, G., (007. ow strength onrete T ross setion RC beams strengthened in shear by NSM tehnique, Workshop-Materiali ed Approi Innovativi per il Progetto in 3

24 Zona Sismia e la Mitigazione della ulnerabilità delle Strutture, University o Salerno, Italy, 1-13 February. Dias, S.J.E. and Barros, J.A.O., (008. Shear Strengthening o T Cross Setion Reinored Conrete Beams by Near Surae Mounted Tehnique, Journal o Composites or Constrution, ASCE, ol. 1, No. 3, pp Monti, G., Renzelli, M., uiani, P., (003 FRP Adhesion to Unraked and Craked Conrete Zones, Proeedings o the 6th International Symposium on Fibre-Reinored Polymer (FRP Reinorement or Conrete Strutures (FRPRCS-6, Singapore, July, Monti, G., Santinelli, F., otta, M.A., (004 Mehanis o FRP Shear Strengthening o RC beams, Pro. ECCM 11, Rhodes, Greee. Monti, G., otta, M.A., (007 Tests and design equations or FRP-strengthening in shear, Constrution and Building Materials (006, 1(4, April 007, Mohammed Ali, M.S., Oehlers, D.J., Seraino, R. (006. ertial shear interation model between external FRP transverse plates and internal stirrups, Engineering Strutures 8, Mohammed Ali, M.S., Oehlers, D.J., Grith, M.C., Seraino, R. (007. Interaial stress traner o near surae-mounted FRP-to-onrete joints, Engineering Strutures 30, Rizzo, A. and De orenzis,., (009 Behaviour and apaity o R beams strengthened in shear with NSM FRP reinorement, Constrution and Building Materials, ol. 3, n. 4, April 009, Sekuli, A., Curnier, A., (006. An original epoxy-stamp on glass-dis speimen exhibiting stable debonding or identiying adhesive properties between glass and epoxy, International Journal o Adhesion and Adhesives, ol. 7, pp Sena-Cruz, J.M., Barros, J.A.O., (004. Bond between near-surae mounted CFRP laminate strips and onrete in strutural strengthening, Journal o Composites or Constrution, ASCE, ol. 8, No. 6, pp Yuan, H., Teng, J.G., Seraino, R., Wu, Z.S., Yao, J. (004. Full-range behavior o FRP-to-onrete bonded joints, Engineering Strutures, 6, Zhai,.., ng, G.P., Wang, Y.W., (008. Eet o nano-al O 3 on adhesion strength o epoxy adhesive and steel, International Journal o Adhesion and Adhesives, ol. 8, No. 1-, pp

25 TABE CAPTIONS Table 1. alues o the parameters haraterizing the beams adopted to appraise the ormulation proposed. Table 1. alues o the parameters haraterizing the beams adopted to appraise the ormulation proposed. Beam abel exp θ β s mm Steel Stirrups,1 kn, kn,3 kn exp kn mm u kn S-3-I F6/ S-5-I S-8-I S-3I45-I S-5I45-I S-8I45-I * S-3I60-I S-5I60-I S-7I60-I S-7-II F6/ S-4I45-II S-7I45-II S-4I60-II S-6I60-II S-7-II F6/ * S-4I45-II S-7I45-II S-4I60-II S-6I60-II S-5I45-III F6/ S-5I45F1-III ** S-5I45F-III ** S-5I45-III F6/ S-5I45F-III ** S-9I45-III F6/ S-9I45-III F6/ S-5I60-III F6/ S-5I60-III F6/ S-5I60F-III ** S-8I60-III F6/ S-8I60-III F6/ S-6-III F6/ S-10-III I beams tested by Dias & Barros (006 and haraterized by a/d equal to.5 and m equal to 31.1 MPa; II beams tested by Dias et al. (007 and haraterized by a/d equal to.5 and m equal to 18.6 MPa; III beams tested by Dias (008 and haraterized by a/d equal to 3.3 and equal to 59.4 MPa. m * beams whose experimental value o NSM shear strength ontribution is aeted by some disturbane; ** beams whih were subjeted to pre-raking. 5

26 FIGURE CAPTIONS Fig. 1. Main physial-mehanial eatures o the theoretial model and alulation proedure: a average-available-bond-length NSM strip and relevant prism o surrounding onrete, b adopted loal bond stress-slip relationship, NSM strip onned to the orresponding prism o surrounding onrete and semi-pyramidal rature surae, d setions o the onrete prism. Fig.. Calulation proedure: main algorithm. Fig. 3. Determination o bond-based onstitutive law ( ; R and bond traner length tr, ( ; R (a invariant distribution o bond shear stress ( x τ and slip ( x out o the three bond phases and relative limits ( ( 1,,3 R : or an innite value o R and (b-g singling i = as a result o the progressive migration o τ ( x rom the oaded End (E towards the Free End (FE or whatever value o R and by inreasing. Fig. 4. Bond-based onstitutive law o a single NSM FRP strip: (a relationship between the bond traner length ( tr, ; R and the imposed end slip or dierent values o resisting bond length R ; (b bi-dimensional and ( three-dimensional representation o the relationship between the ore tranerrable by bond ( ; R and or dierent values o R. Fig. 5. Determination o the omprehensive onstitutive law: low hart. Fig. 6. Single NSM FRP strip omprehensive onstitutive law in the ase in whih onrete rature remains shallow: a resulting onstitutive law ( ; length tr, ( ; in a bi-dimensional representation, b resulting overall traner, setion o the onrete prism and ourrene o subsequent ratures and d resulting onstitutive law ( ; in a three-dimensional representation. Note that this plot has been done or an initial resisting bond length equal to the eetive bond length but this does not aet the generality o the exposition. Fig. 7. Single NSM FRP strip omprehensive onstitutive law in the ase in whih onrete rature is deep: a resulting onstitutive law ( ; ( tr, ; in a bi-dimensional representation, b resulting overall traner length, setion o the onrete prism and ourrene o subsequent ratures and d resulting onstitutive law ( ; in a three-dimensional representation. Note that this plot has been done or an 6