THREE DIMENSIONAL MECHANICAL MODEL TO SIMULATE THE NSM FRP STRIPS SHEAR STRENGTH CONTRIBUTION TO A RC BEAM: PARAMETRIC STUDIES

Transcription

1 THREE DIMENSIONA MECHANICA MODE TO SIMUATE THE NSM FRP STRIPS SHEAR STRENGTH CONTRIBUTION TO A RC BEAM: PARAMETRIC STUDIES Vincenzo Bianco 1, J.A.O. Barros 2 and Giorgio Monti 3 Abstract: This paper presents the results o a parametric study carried out with a a mechanical model recently developed to simulate the shear strength contribution provided by a system o Near Surace Mounted (NSM) Fiber Reinorced Polymer (FRP) strips applied to a Reinorced Concrete (RC) beam. That model, developed ullling equilibrium, kinematic compatibility and constitutive laws o both materials, concrete and FRP, and local bond between themselves, takes into consideration the possibility that the NSM strips may ail due to: loss o bond (debonding), concrete semi-conical tensile racture or strip tensile rupture. The model also takes into consideration: a) interaction between progressive orce transerred by bond to the surrounding concrete and its tensile racture and b) bi-directional interaction among adjacent strips placed on the two sides o the strengthened beam cross-section web. In the rst part o the paper attention is ocused on the bond-based behavior o a single NSM FRP strip mounted on a concrete prism. The inluence o each geometrical-mechanical parameter on the peak orce transerable through bond stresses to the surrounding concrete, excluding the possibility o either concrete racture and strip rupture, is analyzed. In the second part o the paper attention is ocused on the comprehensive behavior o a single NSM FRP strip mounted on a concrete prism. The inluence o each geometrical-mechanical parameter on the peak orce transerable to the surrounding concrete, also including the possibility o both concrete racture and strip rupture, is analyzed. The third part o the paper is devoted at assessing the inluence o each geometrical-mechanical parameter on the imum shear strength contribution provided by a system o NSM FRP strips applied to a RC beam. The results o these studies are presented along with the main ndings. Keywords: Numerical Model; FRP; NSM; Shear Strengthening; Concrete Fracture; Debonding; Tensile Rupture, Parametric Studies. Introduction Shear strengthening o RC beams by NSM technique consists in gluing FRP strips by an adhesive into thin shallow slits cut onto the concrete cover o the beam web lateral aces. This technique has been extensively investigated in recent 1 Post Doctoral Fellow, Dept. o Structural Engrg. and Geotechnics, Sapienza University o Rome, via A. Gramsci 53, 197 Rome, Italy. vincenzo.bianco@uniroma1.it. Corresponding Author. Tel , Fax Associate Proessor, Dept. o Civil Engineering, University o Minho, Campus de Azurém, Guimarães, Portugal. barros@civil.uminho.pt. 3 Full Proessor, Dept. o Structural Engrg. and Geotechnics, Sapienza University o Rome, via A. Gramsci 53, 197 Rome, Italy. giorgio.monti@uniroma1.it. 1

2 years (e.g.: Mohammed Ali et al. 26 and 27, Bianco et al. 26 and 27, Dias et al. 27, Dias and Barros 28, Rizzo and De orenzis 29; Dias and Barros 211a,211b). A three dimensional mechanical model was recently developed to simulate the NSM FRP strips shear strength contribution to a RC beam (Bianco 28, Bianco et al. 29ab). That model was developed ullling equilibrium, kinematic compatibility and constitutive laws o both materials FRP and concrete and bond between themselves. From a physical point o view that model assumes that, during the loading process o a RC beam strengthened in shear by the NSM technique, the strips eectively crossing the Critical Diagonal Crack (CDC) oppose its widening by anchoring to the surrounding concrete to which they transer, through bond stresses, the orce originating at their intersection with the CDC. The relative movement o the two parts into which the CDC divides the beam web, imposes on the strips available resisting bond lengths an increasing end slip (Bianco et al. 29a). As unction o the mechanical and geometrical properties characterizing a certain NSM shear strengthened RC beam, the ailure mode assumed by each strip can be one o the ollowing: a) complete extraction o the NSM FRP strip due to loss o bond throughout the strip available resisting bond length, in case where concrete mechanical properties are relatively high, which is, indeed, a very limit situation (debonding), b) concrete semi-conical racture that reaches the strip ree extremity (concrete semi-conical ailure), c) concrete semi-conical racture that stops progressing midway between loaded and ree ends, with consequent debonding o the remaining portion o the available bond length (mixed-semi-cone-plus-debonding) and d) rupture o the strip independently o an initial concrete racture (strip rupture). Note that the last three ailure modes are brittle, while the rst is more ductile (Bianco et al. 29a). The mechanical model herein applied to carry out parametric studies takes also into account the possibility that strips placed on the two sides o the beam web can interact with each other. The analytical details o the mechanical model are herein omitted, or the sake o brevity, but they can be ound elsewhere (Bianco 28, Bianco et al. 29a-b). The good predictive perormance o this model was appraised on the basis o a large amount o experimental results (Bianco 28). In the rst part o the paper, the model is applied to evaluate the inluence o each geometrical-mechanical parameter on the peak orce that a strip, near surace mounted on a concrete prism and subjected to an increasing imposed end slip, can transer through bond stresses to the surrounding concrete. To that aim, the deormability o the concrete prism is accounted or, while the possibilities o either concrete racture or strip rupture are excluded by assuming the tensile strength o both materials, concrete and FRP, innitely large. In the second part o the paper, the model is applied to evaluate the inluence o each geometrical-mechanical parameter on the ultimate load that a single strip, near surace mounted on a concrete prism and subjected to an increasing imposed end slip, can comprehensively transer to the surrounding concrete. For this purpose, the possibilities o occurrence o either concrete semi-conical tensile racture or strip tensile rupture are accounted or. 2

3 In the third part o the paper, the model is applied to evaluate the inluence o each geometrical-mechanical parameter on the peak shear strength contribution provided by a system o NSM FRP strips to a RC beam. Bond-based behavior o a single NSM FRP strip subject to an imposed end slip Applying the model to the case o a single NSM FRP strip mounted on the surace o a concrete prism (Fig. 1), and neglecting the possibilities o either concrete semi-conical tensile racture and strip rupture occur, it is possible to bd determine the bond-based constitutive law o the strip ( ; i ) V δ. This latter is the curve providing the bond-based orce bd V that the generic i-th strip, with resisting bond length, can transer, through bond stresses, to the surrounding concrete as unction o the value o the increasing imposed end slip δ i (Fig. 1a-b). The bond-based bd constitutive law ( ; i ) V δ o a given NSM FRP strip depends on the ollowing parameters (Bianco et al. 29a-b) (Fig. 1d-e): strip cross section thickness thickness a and width a c and width b c ; concrete deormability b ; strip resisting bond length ; concrete prism cross section E (which is unction o the concrete compressive strength cm ); strip s Young s Modulus E and values o bond stress (, 1, 2 τ τ τ ) and slip ( δ 1, δ 2, δ 3 ) dening the local bond stress-slip relationship (Fig. 1e). The analytical details necessary to evaluate the constitutive law o a given NSM FRP strip are herein omitted, or the sake o brevity, but they can be ound elsewhere (Bianco et al. 29a-b). The constitutive laws ( ; ) bd i V δ o NSM FRP strips o dierent values o can be plotted both in a bi-dimensional ( ;δi ) and in a bd i three-dimensional ( V ; ;δ ) Cartesian orthogonal reerence system. The peak bond-transerred orce bd bd, V increases, by increasing bd,, up to the value ( Re ) V corresponding to the eective resisting bond length Re, which is the value o beyond it any urther increase o the resisting bond length does not yield any urther peak load gain. In a bi-dimensional representation (Fig. 1a), the point ( bd ; i ) V δ representative o the state o the NSM strip o resisting bond length linear horizontal or values o moves, or increasing values o i δ, on the same branch, non-linear ascending or Re or > Re, respectively, common or each value o, as long as bd bd is larger or equal to the value o the necessary bond transer length ( δ ) (Bianco et al. 29a-b). The ( δ ), which is unction o δ i only, is the bond length that would be necessary to entirely transer to the surrounding concrete a orce equal to the one originating in the strip loaded end, due to the imposition o δ i (Bianco et al. 29a). When the bd constitutive laws ( ; i ) V δ o NSM FRP strips o dierent and increasing values o resisting bond length are represented in a three dimensional reerence system, they orm a continuous surace (Fig.1b). tr i tr i 3

4 The parametric study presented in this paragraph concerns the inluence o each parameter on the peak bond-transerred orce or dierent values o. Note that or each parameter was considered a range comprehending values a little beyond those having a strict physical conrmation, in order to assess not only their inluence on the physical behavior o an NSM FRP strip, but also their inluence rom a mere analytical-numerical standpoint (Table 1). The results o the parametric studies are reported in the ollowing. A variation o the concrete Young s Modulus (Fig. 2a). An increase o the FRP Young s Modulus (Fig. 2d). In act, curves providing E c does not yield any variation o bd,, or whatever value o E yields a resisting-bond-length-dependent increase o bd, increases with E, with a rate that is as higher as larger is the value o bd, as unction o E and, or which an increase o E can be better elt since resisting bond length Re ( ) bd,. Moreover, the show two phases: a rst one, coincident or all o the values o is larger than the corresponding value o the eective E and a second one, dierent or all o the values o, in which an increase o E can less be elt since is smaller than Re ( ) E. An increase o the strip cross section dimensions, either a or b, yields a resisting-bond-length-dependent increase o bd, bd,. In particular, or the case o the strip cross section thickness a, the curves providing V as unction o a, or dierent values o, are superimposed as long as it is Re ( a ) and present a dierent increasing rate, as larger is, or values o Re ( a ) <. bd, A variation o the prism cross section dimensions, either a c or b c, does not yield any appreciable variation o, or whatever value o (Fig. 2c,). The peak bond-transerred load bd, increases by increasing the resisting bond length up to a certain value beyond which any urther increase does not produce any urther gain in terms o resistance (Fig. 1c). That threshold value o length, according to the terminology already adopted or the Externally Bonded Reinorcement (EBR), is herein labeled as eective resisting bond length Re is equal to 486 mm. Re. Note that or the values o the mechanical parameters herein adopted (Table 1), As regards the inluence o the parameters characterizing the local bond stress-slip relationship, it arises that marginally aected when a realistic range o τ values is considered, or whatever value o bond-transerred orce 4 bd, V is (Fig. 3a). The peak bd, V increases with a resisting-bond-length-dependent-rate by increasing the value o either

5 τ 1 or τ 2. In particular, bd, V increases by increasing τ 1 with a rate that is larger as shorter is the value o bd, V increases with τ 2 with a rate that is higher as larger is the value o (Fig. 3b-c). increasing values o δ 1 with a resisting-bond-length-dependent-rate: in act, the larger the value o the rate with which bd, V decreases or increasing values o δ 1 (Fig. 3d). An increase o either δ 2 or δ 3 (Fig. 3e-) yields an increase o dependent. In particular, or the case o δ 3, the curves providing, while bd, V decreases or, the smaller is bd, V with a rate that is resisting bond length bd, V as unction o δ 3, or the dierent values o, are superimposed, presenting the same trend, as long as is larger or equal to Re ( δ 3 ). Note that, or all o the parameters analyzed, the curves providing various values o bd, V as unction o the generic parameter or the tend, by increasing the value o, to an envelope curve. Moreover, or values o larger than a certain value, the curves providing envelope one. bd, V as unction o the generic parameter are all superimposed to the Comprehensive behavior o a single NSM FRP strip subject to an imposed end slip The model was also applied to study the comprehensive behavior o a single NSM FRP strip, mounted on a concrete prism and subjected to an increasing imposed end slip. For this purpose, the possibilities o both concrete semi-conical tensile racture and strip tensile rupture were also taken into account. Typically, according to the model herein adopted (Bianco 28, Bianco et al. 29a), increasing the imposed end slip to the single NSM FRP strip, co-axial and progressively larger semi-conical concrete ractures subsequently orm around the strip since the very rst load steps (Fig. 4b,c). Contextually at the occurrence o each concrete semi-conical racture, the initial value o the resisting bond length progressively reduces. Moreover, due to the ormation o successive co-axial semi-conical racture suraces around the strip, the resulting concrete racture envelope surace, which corresponds to the last and larger semi-conical surace, starting rom the loaded end, progressively penetrates within the concrete prism (Fig. 4c). It can happen that the vertex o the last semi-conical concrete racture surace places midway between the loaded and ree end or reaches this latter. Moreover, contextually at the occurrence o each o those concrete ractures, the point representative o the state o the strip, with coordinates ( ; ; i) n ( ; ) bd i V δ o the previous value o resisting bond length V δ leaps, or a constant value o δ i, rom the bond-based constitutive law n to the one corresponding to the new value o resisting bond length n 1 + (Fig. 5). The leap o the point ( ; ; i) V δ representative o the state o the strip rom one bond- 5

6 based constitutive law to the other is only visible in a three dimensional representation, as long as the concrete racture is supercial (Fig. 5c,e). Where supercial concrete racture means that, each time the point representative o the state o the strip leaps rom one bond-based constitutive law to another, the updated value o bd necessary bond transer length ( ) δ (Fig. 5c,e). On the contrary, the leap o the point ( ; ; i) tr i is larger or equal to the V δ representative o the state o the strip rom one bond-based constitutive law to the other is also visible in a two dimensional representation, as long as the concrete racture is deep (Fig. 5a,d). Where deep concrete racture means that, when the point representative o the state o the strip leaps to the last bond-based constitutive law, the new value o is bd smaller than the necessary bond transer length ( δ ) corresponding to the current value o δ i (Fig. 5a,d). When the point ( ; ; i) tr i V δ representative o the state o the strip eventually leaps on the bond-based constitutive law o a strip with null resisting bond length, the ultimate conguration is characterized by a semi-conical concrete racture whose vertex coincides with the strip ree end. It can also happen that, ater an initial semi-conical concrete racture, at a certain point o the loading process, the strip ruptures and the point ( ; ; i) strip abruptly alls on the plane ( ; i) δ and the relevant orce annuls (Fig. 5c,). V δ representative o the state o the Note that, whatever the ailure mode characterizing the specic single NSM FRP strip, due to the continuity characterizing the surace envelope o the bond-based constitutive laws, it always exists an equivalent value o the resisting bond length eq, which is the value o the resisting bond length to which corresponds a bond-based bd eq constitutive law ( ;δ i ) constitutive law ( ; i ) V whose peak value bd, is equal to the peak value V δ o the initial value o the resisting bond length (Fig. 5). o the comprehensive Ultimately, the comprehensive constitutive law o a single NSM FRP strip mounted on a concrete prism can be one o the ollowing types (Fig. 4a): (1) either rupture o the strip, preceded or not by a supercial semi-conical racture, or deep concrete semi-conical racture up to the ree end; (2) deep concrete semi-conical racture with the concrete racture stopping midway between loaded and ree end; supercial concrete racture with the last value assumed by (3), equal (4) or larger (5) than the eective resisting bond length. shorter In the ollowing o this paragraph, a parametric study, regarding the inluence o each o the geometrical-mechanical parameters on the peak value o the comprehensive constitutive law ( ; i ) V δ o a single NSM FRP strip, is presented. Note that even in this case, each parameter was varied within a range comprehending values a little beyond those having a strict physical conrmation, in order to asses not only their inluence on the physical behavior o an NSM FRP strip, but also their inluence rom a mere analytical-numerical standpoint. 6

7 Besides the parameters already introduced in the previous paragraph, the comprehensive behavior o a single NSM FRP strip also depends on (Table 1): strip tensile strength u, angle between axis and generatrices o the concrete semiconical racture surace α (Fig. 4b,c). Among the derived parameters, the concrete average tensile strength ctm, unction o cm is, in the present work, obtained as unction o the concrete compressive strength by means o the ormulae reported by the CEB Fip Model Code 9. The curves representing the variation o elastic-perectly-plastic behavior or whatever value o with respect to the FRP strip tensile strength u present a bi-linear (Fig. 6a). Along the rst linear branch, the ailure mode is rupture o the strip, due to the reduced value o the strip tensile strength tr V ( a b ). Along the second linear u branch, the ailure mode is governed by supercial concrete racture or the shorter values o the initial resisting bond length and by deep concrete racture or the larger values o the initial resisting bond length. Note that, since the value o u does not aect the pure bond-based behavior o a NSM FRP strip, or a given value o and or values o u larger than the one in correspondence o which the strip no longer ruptures, the ultimate conguration and the relevant peak load V no longer change by increasing u. Note also that the curves providing the values o as unction o u or the various values o tend to a limit curve which denes the envelope o the various curves. In act, or the values herein assumed or the other parameters, the curve providing same regardless o the value assumed by The curves ( ; ) V E providing or values o larger than 4 mm (Fig. 6a). as unction o u is exactly the as unction o the strip Young s Modulus E present a pseudo bi-linear trend or whatever value o. Along the rst branch, the ultimate behavior is characterized by supercial concrete racture and, since the equivalent values o the resisting bond length are larger or equal to the corresponding values o eq the eective resisting bond length ( ( ) Re ( ) ound (Fig. 6d). Nevertheless, due to the reduction o value o, or a given value o E E ), the same trend characterizing the pure-bond behavior can be pseudo horizontal branches o the curves ( ; ) ascribed to the supercial concrete racture, the imum, is smaller than the imum o bd, or the same value o V E correspond, or the shorter values o. The ( < 35 mm ), to a deep concrete racture ailure mode and, or the longer values o ( 35 mm ), to the rupture o the strips. The horizontal branches o the curves ( ; ) V E tend, or increasing values o, to the limit curve corresponding to 7

8 the value o the strip rupture capacity V 42. kn tr =. Note that, on the horizontal branches o the curves V ( E ; ) or the shorter values o, despite a luctuation, due to concrete racture, around an average value, is almost constant even i, by increasing This is due to the act that, by increasing value o bd, reduce (Bianco 28). The curves V ( a ; ) and ( ; ) E, concrete racture deepens and eq progressively becomes a smaller aliquot o. E, the values o the resisting bond lengths necessary to transer the same V b providing the value o as unction o the strip cross section dimensions present, or whatever value o, a pseudo bi-linear trend (Fig. 6b,e). Along the rst branch o those curves, due to the reduced area o the strip cross section, the ailure mode is rupture o the strip and linearly or increasing values o either increases a or b. Along the second branch o those curves, the ultimate conguration is o the mixed-semi-cone-plus-debonding type and, or the shorter values o, the ailure mode is supercial concrete racture while or the longer values o, the ailure mode is deep concrete racture. The horizontal branches o those curves tend, or increasing values o, to the limit curve corresponding to the value o approximately V = 62. kn. From a certain value o on ( 35 mm or a and 4 mm or b ), the curves above overlap on the limit curve and the imum value o The curves V ( ac; ) and ( c; ) V b providing the value o does not urther increase or increasing values o. as unction o the concrete prism cross section dimensions present, or whatever value o, a pseudo bi-linear trend (Fig. 6c,). Along the rst branch o those curves, since the prism cross section is small, the successive co-axial semi-conical racture suraces, whose axis is a progressively larger amount o, soon intersect the concrete prism aces (Fig. 4b,c). Thus, the concrete semiconical racture capacity c, obtained integrating the concrete average tensile strength throughout the surace resulting rom the intersection o the right semi-conical surace with the prism aces, is very low (Bianco et al. 29a-b). Due to the small value o the concrete racture capacity, along the rst branch o the curves V ( ac; ) and ( c; ) V b, the ailure mode is very deep concrete racture with the vertex o the last semi-conical racture surace reaching the strip ree end (Fig. 4b,c). Along the second branch o those curves, or increasing values o either a c or b c, and or the shorter values o, concrete racture becomes, as long as the largest semi-conical surace intersects the prism walls, progressively more supercial. From the value o either a c or b c, in correspondence o which the largest semi-conical 8

9 surace no longer intersects the prism aces, the ultimate conguration and the corresponding value o, or a given value o, remains practically unchanged. Along the second branch o those curves, or values o larger than a certain value ( 3 mm or a c and 25 mm or b c ), the ailure mode is governed by the strip rupture and those branches overlap on the limit curve corresponding to the value o V = V = 42. kn (Fig. 6c,). The curve ( ) V providing the value o as unction o the value o tr also presents a pseudo bi-linear trend (Fig. 7c). Along the rst branch, the ailure mode is concrete semi-conical racture, either supercial or deep, with an ultimate conguration o the mixed-semi-cone-plus-debonding type. Along the second branch, the ailure mode is rupture o the strip. Note that even in this case as or the pure bond behavior, it is possible to single out an eective value value o, which can be labeled as comprehensive eective resisting bond length Re, beyond which any urther increase o does not produce any urther gain in terms o. In general, the comprehensive eective resisting bond length Re, due to the occurrence o one o either concrete racture or strip rupture, is shorter than the eective resisting bond length Re (see also Fig. ). The curves V ( α ; ) and ( cm; ) V providing the value o dening the concrete mechanical properties, either cm or α, present, or whatever value o as unction o either o the parameters, a nonlinear trend in which, in the most general case, three successive branches can be singled out (Fig. 7a,b). Along the rst branch, or small values o either cm or α, since concrete mechanical properties are very low, concrete racture is so deep to reach the strip ree extremity. This rst branch o the curve is continuous since, going rom one value to the other o the parameter analyzed ( cm or α ), the peak value is always equal to the concrete semi-conical racture capacity c associated to the imum semi-conical surace that can orm inside the concrete prism. Along this rst branch, c the concrete racture capacity ( ) or an increase o either ( ) V, integral o ctm on the imum semi-conical surace varies with continuity c ctm cm or α. In this latter case, ( ) V varies with continuity i either the imum semi-conical surace intersects (or larger values o α ) or not (or smaller values o α ) the concrete prism aces (Fig. 4). By increasing the values o either cm or α, concrete racture progressively becomes more supercial and the equivalent value o the resisting bond length becomes a progressively larger aliquot o. Along the second branch o the curves V ( α ; ) and ( cm; ) V the ultimate conguration is o the mixed-semi-cone-plus-debonding 9

10 type. Such second branch is characterized by a two-old luctuation o the values, by varying either o the parameters analyzed ( cm or α ) and or a given value o, around the values corresponding to an average and continuous curve: a micro-luctuation ad a macro-luctuation. The ormer is a luctuation that has a numerical origin and is due to the way in which the concrete semi-conical racture phenomenon was modeled (Bianco 28). The latter, which consists in the presence o some steps upwards, is due to the act that, or increasing values o the concrete mechanical properties, the number o successive reductions o (Fig. 4), contextual to the occurrence o successive concrete semi-conical ractures, reduces or increasing values o either o the parameters analyzed ( cm or α ). Along the third branch, attained indeed only or the larger values o and or increasing values o the concrete mechanical properties, the ultimate conguration is composed o a supercial semi-conical concrete racture ollowed by the rupture o the strip itsel. Such third branch is thereore horizontal and equal to the value o the strip rupture capacity that, or the values herein assumed or the other parameters, is equal to 42. kn. The curves V ( ; ) α and ( ; ) cm V, or increasing values o, progressively become closer to each other up to overlapping on a limit curve or values o larger than a certain limit value that, or the values herein assumed or the other parameters, is equal to 45 mm or α and to 3 mm or cm. Note that the curves ( cm; ) V start rom a value o cm equal to 8. MPa. This is due to the act that the concrete average tensile strength ctm was calculated rom the concrete average compressive strength by means o the ormulae reported by the CEB-FIP Model Code 9 that provide values o ctm larger than zero or values o cm larger than 8. MPa. Note also that, i either cm or α were increased innitely, or a given value o, the value o would tend to the corresponding bond-based peak load ( ) bd, V, as long as this latter is smaller than the strip rupture capacity. The peak orce that an NSM FRP strip o a given resisting bond length can resist does not vary by increasing the value o the parameter τ, characterizing the adopted local bond stress-slip relationship, or whatever value o the initial resisting bond length (Fig. 8a). Actually, the parameter τ was already ound not to aect the peak bondtranserred orce bd, (Fig. 3a). Nonetheless, due to the occurrence o other phenomena such as either concrete tensile racture or strip rupture, the peak value o the curve V ( ; ) τ is lower than the peak value o the bd, corresponding curve V ( ; ) τ or a value o =. For the shorter values o, the ailure mode is 1

11 governed by concrete tensile racture, either supercial or deep while, or values o value, the ailure mode is the strip rupture. larger or equal to a certain The values o the parameters τ 1 and τ 2 have a negligible inluence on the peak value o the comprehensive constitutive law o a NSM FRP strip o a given resisting bond length (Fig. 8b,c). In act, the curves V ( 1 ; ) τ and V ( 2 ; ) τ do not show appreciable variations, by increasing either τ 1 or τ 2, or whatever value o. For the shorter values o, the ailure mode is governed by concrete tensile racture, either supercial or deep while, or values o larger or equal to a certain value, the ailure mode is the strip rupture. Each o the bond slip values characterizing the local bond stress-slip relationship, either δ 1 or δ 2 or δ 3, has a marginal inluence on the peak value o the comprehensive constitutive law o a NSM FRP strip o a given value o the initial resisting bond length (Fig. 8d-). For the shorter values o racture, either supercial or deep while, or values o strip rupture. Note that the curves providing, the ailure mode is governed by concrete tensile larger or equal to a certain value, the ailure mode is the as unction o either o the various parameters dening the local bond stress-slip relationship tend, or increasing values o, to overlap to the limit curve corresponding to the strip rupture V tr = V. NSM FRP strips shear strength contribution to a RC beam The mechanical model adopted in the previous paragraphs, when applied aithully to its original eatures (Bianco 28, Bianco et al. 29a), allows the shear strength contribution provided by a system o NSM FRP strips to a RC beam to be evaluated as unction o the CDC opening angle γ (Fig. 9). The relation V ( γ ) may be evaluated or whatever relative geometrical position that the occurred CDC should assume with respect to the system o NSM FRP strips. Nonetheless, in the present work, three geometrical congurations only ( k = 1, 2, 3 ) were taken into consideration: ( k = 1 ) the minimum number o strips that can eectively cross the CDC with the rst one placed at a distance equal to V the strips spacing s rom the CDC origin; ( k = 2 ) an even number o strips symmetrically placed with respect to the crack axis; ( k = 3 ) an odd number o strips with the central one attaining the imum available bond length by being located along the crack axis (Bianco 28). To evaluate the relationship V ( γ ) it is necessary to introduce, besides the input parameters already introduced in the previous paragraphs, also the ollowing ones (Figure 9): the beam web cross section depth h w and width b w, the CDC inclination angle θ with respect to the beam longitudinal axis, the strips 11

12 inclination angle with respect to the beam longitudinal axis β and the CDC opening angle increment γɺ. The typical relationship V ( γ ), sum o the contribution o the strips eectively crossing the CDC, is characterized by abrupt decays o V that correspond to the ailure o the various strips (Fig. 9c-). Typically, according to that model, during the loading process o a RC beam strengthened in shear, co-axial semi-conical concrete ractures orm around each o the strips eectively crossing the CDC since the rst load steps. Due to the ormation o successive co-axial semiconical racture suraces around each o the strips, the resulting concrete racture envelope surace, starting rom the CDC, progressively penetrates within the web core on either side o the CDC plane (Bianco et al. 29a). The mechanical model mentioned above, was applied to evaluate the inluence o each o the input parameters (Table 2) on the peak NSM FRP strips shear strength contribution to a RC beam V and the obtained results are presented below. The Reerence Beam (RB) assumed or the parametric studies along with the range o values assumed or each parameter, are listed in Table 2. The values o the parameters characterizing the RB have been chosen in such a way to result approximately the average values o those that can be met in real practice. In some cases, the specic parameter is varied within a range that comprehends values a little beyond those having a strict physical conrmation, in order to assess not only their inluence on the physical behavior o RC beams strengthened in shear by the NSM technique, but also their inluence rom a mere analytical-numerical standpoint. The range o variation o cm was limited to the values 1-9 MPa (Table 2) or which concrete can be considered as structural concrete, in accordance to the international regulations (e.g. CEB-FIP Model Code 199). The load step γɺ slightly inluences, or the range o values in which it was herein varied ( ), the peak NSM shear strength contribution V (Fig. 1a). In this scenario, it is deemed reasonable to assume a value o the load step equal to ɺ γ =.1 since it guarantees a good compromise between accuracy o prediction and computational demand. The peak NSM shear strength contribution V decreases by assuming increasing values o the CDC inclination angle θ since, other parameters being the same, the number o strips eectively crossing the CDC decreases (Fig. 1b). The peak NSM shear strength contribution increases linearly or increasing value o the web cross section depth V, or the range o values herein assumed or the other parameters, h w (Fig. 1c), since both the number o strips eectively crossing the CDC and their available bond lengths increase in turn. Actually, up to a certain value o h w, the linear increase o V is primarily due to the increase o the strips available bond lengths while, or larger values o 12

13 h w, since the strips available bond lengths start to be larger than the comprehensive eective resisting bond length (Fig. 7c), the linear increase o V is due to the increase in the number o strips eectively crossing the CDC. By increasing the RC beam web cross section width b w (Fig. 1d), the peak NSM shear strength contribution presents a two-phases behavior with an almost clear-cut division between themselves in correspondence o a value o approximately b = 15 cm. For values o the web width b 15 cm w w, by increasing b w, V V increases almost linearly up to a imum while, or values o the web width bw > 15 cm, any urther increase o b w does not yield any urther increase o V. This is due to the act that, or b 15 cm, the strips symmetrically placed on the opposite sides o w the strengthened web interact transversally with each other while, or values o b > 15 cm, due to the lack o interaction between strips placed on the opposite sides o the beam web, any urther increase o b w does not produce any change in the overall response. w The peak NSM shear strength contribution V presents, with respect to either o the CFRP strip s cross-section dimensions ( a or b ), a three-phases trend (Fig. 1e-). In the rst phase ( a 1.5 mm or b 8 mm ), the prevalent ailure mode is tensile rupture o the strips and V increases almost linearly by increasing either a or b. In act, ater a soon and shallow concrete racture, most o the strips crossing the CDC rupture due to their reduced crosssection area. In the intermediate stretch ( 1.5 < a 4.5 mm or 8 < b 12 mm ) V increases, by increasing either a or b, by a decreasing rate since the number o strips undergoing rupture progressively reduces and the number o strips ailing or concrete racture, either supercial or deep and up to the ree end, increases. Along the third stretch ( a > 4.5 mm or b > 12 mm ), all o the strips eectively crossing the CDC ail or either deep or supercial concrete racture and their contribution to the overall shear strength contribution V no longer depends on either a or b (Fig. 6). However, or very large values o either o the strip cross-section dimensions, the model progressively loses accuracy underestimating the concrete racture capacity since the approximation o the concrete racture surace as a semi-cone whose axis lies on the web ace (Fig. 4) may no longer be acceptable (Bianco et al. 29a). By increasing the spacing s between adjacent strips, even though the contribution provided by the single strip would increase as a consequence o the increase o the cross section o the relevant concrete prism (Figs. 9a and 4b-c), the overall contribution to the peak NSM shear strength V signicantly decreases (Fig. 11a) since the number o strips eectively crossing the CDC decreases in turn. For the third geometrical conguration ( k = 3 ), or which, rom a 13

14 certain value o s on, the number o strips eectively crossing the CDC remains unchanged and equal to one central strip only, the same plastic stretch, already highlighted or the single strip (Fig. 6c) can be ound again. By increasing the value o the inclination o the strips with respect to the beam axis β, the peak NSM shear strength contribution decreases (Fig. 11b) mainly because, other parameters being the same, both the number o strips eectively crossing the CDC and their initial available bond lengths decrease. The strip's Young s Modulus E has, on the whole and or the values assumed in the present study ( E = 1 25GPa ), a negligible inluence on V (Fig. 11c). In act, or the range o values o E analyzed, already the single strip peak contribution does not vary or increasing values o E, or each value o and whatever the ultimate conguration (Fig. 6d). Thus, even the overall shear strength contribution V does not vary or increasing values o E, within that range. The trend o the peak NSM shear strength contribution V with respect to the strip tensile strength u shows two phases separated by an almost clear-cut turning point approximately in correspondence o the value o 3. GPa (Fig. 11d). For values o u smaller than 3. GPa, the ultimate behavior is governed by the prevalent rupture o the strips. For values o u equal to or larger than 3. GPa, the ultimate behavior is governed by concrete racture and, whatever the depth o concrete racture penetration, each strip s peak contribution does not change or increasing values o u, whatever their (Fig. 6a). The variation o V with respect to the concrete mechanical properties, either cm or α, presents a non-linear trend in which three successive stretches can be singled out (Fig. 11e-). For values o α 3º and 3 MPa, the prevailing ailure mode o most o the strips eectively crossing the CDC is concrete tensile racture reaching the strips ree ends (Fig. 7a-b). Along the second stretch, or values o 3º < α 4º and 3 < 6 MPa, by progressively increasing the concrete mechanical properties, either cm or α, concrete racture progressively becomes shallower (Fig. 4c) and V increases by a decreasing rate. In act, along the second stretch, since the curve V ( α ) cm cm or V ( ) are characterized by a cm -dependent rate (Fig. 7a-b), and V is equal to the sum o the contribution o strips with dierent values o, V increases by a decreasing rate. Along the third stretch, or values o α > 4º and cm > 6 MPa, V does not urther vary or increasing values o either cm or α. In act, or values o 14

15 α > 4º and > 6 MPa, the peak o the single strips contribution cm ailure mode (Fig. 7a-b). does not vary, independently o the As already ound out or a single NSM FRP strip subjected to an increasing imposed end slip (Fig. 8), the values o the parameters dening the local bond stress-slip relationship do not aect, or the values o the other parameters herein assumed, the peak NSM FRP strips shear strength contribution (Fig. 12). In act, due to the occurrence o other phenomena such as either concrete semi-conical tensile racture or strip rupture, the peak bond-transerred orce bd, V that the initial values o the resisting bond length would be capable o transerring, can never be attained. Conclusions A mechanical model, recently developed to evaluate the NSM FRP strips shear strength contribution to a RC beam throughout its loading process, was herein applied to carry out parametric studies. These latter encompass the studies o the inluence o each geometrical-mechanical parameter on: 1) the peak load that a single NSM FRP strip can transer, through bond stresses, to the surrounding concrete, neglecting both concrete semi-conical racture and strip rupture; 2) the peak load that a single NSM FRP strip can comprehensively transer, taking also into account the possible occurrence o either concrete racture or strip rupture; 3) the peak shear strength contribution that a group o NSM FRP strips can provide to a RC beam. The bond-based constitutive law o a single NSM FRP strip, which is the curve providing the orce that a strip with a given value o resisting bond length can transer, through bond stresses, as unction o the imposed end slip, orms a continuous surace when plotted in a three-dimensional graph or increasing values o resisting bond length. The peak bond-transerred orce, which is the peak value o the strip s constitutive law, increases, or increasing values o the resisting bond length, only up to a certain threshold, labeled eective resisting bond length, beyond which any urther length increase does not produce any urther orce gain. The peak bond-transerred orce increases, with a rate that is as larger as larger is the value o the resisting bond length, or increasing values o either the strip s Young s Modulus or the strip s cross section dimensions. The peak bond-transerred orce increases also, with a rate that is as larger as larger is the value o the resisting bond length, or increasing values o most o the parameters dening the local bond-stress slip relationship. The comprehensive behavior o a single NSM FRP strip subjected to an increasing imposed end slip is characterized by the ormation o successive co-axial semi-conical concrete ractures which can stop progressing midway between loaded and ree end, maybe ollowed by strip rupture i the strip rupture capacity is attained, or progress up to the ree end. Contextually at the occurrence o each concrete racture, the initial value o the resisting bond length progressively 15

16 reduces and the point describing the strip s comprehensive constitutive law in a three dimensional representation leaps rom one bond-based constitutive law to another. I either the concrete racture reaches the strip s ree end or the strip ruptures, the strip s comprehensive constitutive law abruptly annuls. Since the bond-based constitutive law o an NSM FRP strip orms a continuous surace or increasing values o the resisting bond length, it is always possible to single out an equivalent value o the initial resisting bond length whose bond-based constitutive law has a peak value equal to the peak load o the comprehensive constitutive law o the initial resisting bond length. The dependence o the peak value o the comprehensive constitutive law on most o the input parameters shows a nonlinear, initial resisting-bond-length-value-dependent and pseudo elasto-plastic trend. The non-linearity is due to the act that, or a given value o initial resisting bond length, or increasing values o the studied parameter, both the ailure mode and the ultimate conguration attained by the strip progressively change. The peak value o the comprehensive constitutive law increases by increasing the initial value o the resisting bond length up to a certain value, labeled eective value o the initial resisting bond length, beyond which any urther length increase does not yield any urther orce gain. The inluence o each one o the parameters dening the local bond stress-slip relationship on the peak value o the comprehensive constitutive law o a single NSM FRP strip with a given value o initial resisting bond length is negligible. In act, the premature occurrence o either concrete semi-conical racture or strip rupture does not allow to attain the peak bond-transerred orce that a single strip, with the given value o initial resisting bond length, would be capable to attain. The behavior o a system o NSM FRP strips contributing to the shear strength o a RC beam is extremely complex since the strips eectively contributing to the beam shear strength, intersecting the CDC plane, are: 1) characterized by dierent values o the initial resisting bond length, 2) not necessarily orthogonal to the CDC, 3) subjected to dierent values o imposed end slip and 4) bi-directionally interact between each other. The curve providing the NSM FRP strips shear strength contribution as unction o the CDC opening angle is characterized by abrupt orce reductions due to the strips ailure, whatever the ailure mode they singly undergo. The dependence o the peak NSM shear strength contribution on most o the input geometrical-mechanical parameters is extremely non-linear. As already outlined or the comprehensive behavior o a single NSM FRP strip, a variation o each o the parameters dening the local bond stress-slip relationship yields a negligible variation o the peak NSM shear strength contribution, due to the premature occurrence o either concrete semi-conical racture or strip rupture. Acknowledgements 16

17 The authors o the present work wish to acknowledge the support provided by the Empreiteiros Casais, S&P, degussa Portugal, and Secil (Unibetão, Braga). The study reported in this paper orms a part o the research program CUTINEMO supported by FCT, PTDC/ECM/7399/26. Also, this work was carried out under the auspices o the Italian DPC-Reuis Project (repertory n. 54), Research ine 8, whose nancial support is greatly appreciated. 17

18 Reerences Bianco, V., Barros, J.A.O., Monti, G., (26). Shear Strengthening o RC beams by means o NSM laminates: experimental evidence and predictive models, Technical report 6-DEC/E-18, Dep. Civil Eng., School Eng. University o Minho, Guimarães- Portugal. Bianco, V., Barros, J.A.O., Monti, G., (27). Shear Strengthening o RC beams by means o NSM strips: a proposal or modeling debonding, Technical report 7-DEC/E-29, Dep. Civil Eng., School Eng. University o Minho, Guimarães- Portugal. Bianco, V., (28). Shear Strengthening o RC beams by means o NSM FRP strips: experimental evidence and analytical modeling, PhD Thesis, Dept. o Structural Engrg. and Geotechnics, Sapienza University o Rome, Italy, submitted on December 28. Bianco, V., Barros, J.A.O., Monti, G., (29a). Three dimensional mechanical model or simulating the NSM FRP strips shear strength contribution to RC beams, Engineering Structures, 31(4), April 29, Bianco, V., Barros, J.A.O., Monti, G., (29b). Bond Model o NSM FRP strips in the context o the Shear Strengthening o RC beams, ASCE Journal o Structural Engineering, 135(6), June 29. CEB-FIP Model Code 9, (1993) Bulletin d Inormation N 213/214, Final version printed by Th. Telord, ondon, (1993; ISBN ; 46 pages). Dias, S.J.E.; Barros, J.A.O., Shear strengthening o RC T-section beams with low concrete using NSM CFRP laminates, Cement and Concrete Composites Journal, doi:1.116/j.cemconcomp , 211b. Dias, S.J.E.; Barros, J.A.O., Experimental behaviour o RC beams shear strengthened with NSM CFRP laminates, Strain International Journal, doi: /j x, 211a. Dias, S.J.E., Bianco, V., Barros, J.A.O., Monti, G., (27). ow strength concrete T cross section RC beams strengthened in shear by NSM technique, Workshop-Materiali ed Approcci Innovativi per il Progetto in Zona Sismica e la Mitigazione della Vulnerabilità delle Strutture, University o Salerno, Italy, February. Dias, S.J.E. and Barros, J.A.O., (28). Shear Strengthening o T Cross Section Reinorced Concrete Beams by Near Surace Mounted Technique, Journal o Composites or Construction, ASCE, Vol. 12, No. 3, pp Mohammed Ali, M.S., Oehlers, D.J., Seracino, R. (26). Vertical shear interaction model between external FRP transverse plates and internal stirrups, Engineering Structures 28, Mohammed Ali, M.S., Oehlers, D.J., Grith, M.C., Seracino, R. (27). Interacial stress transer o near surace-mounted FRP-to-concrete joints, Engineering Structures 3, Rizzo, A. and De orenzis,., (29) Behaviour and capacity o Rc beams strengthened in shear with NSM FRP reinorcement, Construction and Building Materials, Vol. 3, n. 4, April 29,

19 Sena-Cruz, J.M. (24). Strengthening o concrete structures with near-surace mounted CFRP laminate strips PhD Thesis, Department o Civil Engineering, University o Minho, Guimarães-Portugal. Sena-Cruz, J.M., Barros, J.A.O., (24). Bond between near-surace mounted CFRP laminate strips and concrete, Journal o Composites or Construction, ASCE, Vol. 8, No. 6, pp Yuan, H., Teng, J.G., Seracino, R., Wu, Z.S., Yao, J. (24). Full-range behavior o FRP-to-concrete bonded joints, Engineering Structures, 26,

20 TABE CAPTIONS Table 1 Values or the input parameters adopted or the parametric studies regarding the behavior o a single NSM FRP strip. Table 2 Values or the input parameters adopted or the parametric studies regarding the peak shear strength contribution provided by a system o NSM FRP strips to a RC beam. Table 1 Values or the input parameters adopted or the parametric studies regarding the behavior o a single NSM FRP strip. E u cm α a b a c b c τ τ 1 τ 2 δ 1 δ 2 δ 3 GPa GPa MPa º mm mm cm cm cm MPa MPa MPa mm mm mm Reerence Strip Range o variation Table 2 Values or the input parameters adopted or the parametric studies regarding the peak shear strength contribution provided by a system o NSM FRP strips to a RC beam. γɺ θ h w cm b w cm a mm b mm s cm β E GPa u GPa cm MPa α τ MPa τ 1 MPa τ 2 MPa δ 1 mm δ 2 mm δ 3 mm Reerence Beam Range o Variation

21 FIGURE CAPTIONS Fig. 1. Bond-based behavior o a single Near Surace Mounted (NSM) FRP strip on a concrete prism: a-b) constitutive bd law ( ;δli ) V both in a bi-dimensional and in a three dimensional representation; c) dependence o the imum bond-transerred orce adopted local bond stress-slip relationship. bd, on the resisting bond length ; d) concrete prism with a single NSM FRP strip and e) Fig. 2. Variation o the peak bond-transerred orce by a single NSM FRP strip as unction o: a) concrete Young s Modulus E c, b) strip thickness concrete prism width. a, c) concrete prism thickness a c, d) FRP Young s Modulus Fig. 3. Inluence o the parameters dening the local bond stress-slip relationship on the peak orce single NSM FRP strip can transer, through bond stresses, to the surrounding concrete. E, e) strip width and ) bd, V that a Fig. 4. Comprehensive behavior o a single NSM FRP strip: a) possible comprehensive constitutive law types, b) successive and co-axial semi-conical racture suraces occurring around the NSM strip and c) progressive reduction o the resisting bond length and penetration o the semi-conical racture within the concrete prism. Fig. 5. Comprehensive constitutive law ( ; i ) V δ o a single NSM FRP strip: a,d) deep concrete racture, b,e) supercial concrete racture and c,) strip rupture ater a supercial concrete racture. Fig. 6. Variation o the peak orce that a single strip can comprehensively transer to the surrounding concrete ( ) V, as unction o: FRP s a) tensile strength u and d) Young s Modulus E, strip cross section b) thickness E and e) width E and concrete prism cross section c) thickness a c and ) width b c. Fig. 7. Variation o the peak orce that a single strip can comprehensively transer to the surrounding concrete ( ) V, as unction o: a) the angle between axis and generatrices o the semi-conical concrete racture surace α, b) concrete average compressive strength cm and c) initial resisting bond length Fig. 8. Variation o the peak orce that a single strip can comprehensively transer to the surrounding concrete ( ). V, as unction o the parameters dening the adopted local bond stress-slip relationship. Fig. 9. NSM FRP strips shear strength contribution to a RC beam: (a-b) schematic representation o the beam web shear-strengthened by NSM FRP strips, (c-d) NSM shear strength contribution to the adopted Reerence Beam or two values o the CDC inclination angle and (e-) or two values o the beam web depth. Fig. 1. Inluence on the peak NSM shear strength contribution V o: a) load step γɺ ; b) CDC inclination angle θ ; c) beam cross section depth h w and d) width b w ; e) strip cross section thickness a and ) width b. 21