DESIGN FORMULA TO EVALUATE THE NSM FRP STRIPS SHEAR STRENGTH CONTRIBUTION TO A RC BEAM

Transcription

1 DESGN FORMULA TO EALUATE THE NSM FRP STRPS SHEAR STRENGTH CONTRBUTON TO A RC BEAM incenzo Bianco 1, Giorgio Monti 2 and J.A.O. Barros 3 Abstract: This paper presents the closing step o a synthesis process aiming at deriving, rom a previously developed more complex model, a simple design ormula to evaluate the shear strength contribution provided by a system o Near Surace Mounted (NSM) Fiber Reinorced Polymer (FRP) strips to a Reinorced Concrete (RC) beam. The sel-contained and ready-to-implement set o analytical uations and logical operations is presented along with the main underlying physical-mechanical principles and assumptions. The ormulation proposed is appraised against some o the most recent erimental results and its predictions are also compared with those obtained by the two previous and more sophisticated versions o the same modeling strategy. Monte Carlo simulations are carried out in order to appraise the sensitivity o the NSM shear strength contribution prediction to the value assumed by the input parameters. Keywords: Shear; Strengthening; FRP; C. Analytical modeling; Monte Carlo simulations. ntroduction Shear strengthening o RC beams by NSM technique consists o gluing FRP strips by a structural adhesive into thin shallow slits cut onto the concrete cover o the beam web lateral aces. A comprehensive three-dimensional mechanical model to predict the NSM FRP strips shear strength contribution to a RC beam was recently proposed (Bianco 2008, Bianco et al. 2009a-b and 2010). That model was developed ulilling uilibrium, kinematic compatibility and constitutive laws o both the materials involved (FRP and concrete), as well as the local bond between themselves. Despite its consistency with erimental recordings, that model turned out to be somehow cumbersome to be easily implemented and accepted by proessional structural engineers. More recently, a simpler version o that model was derived rom the more complex one by introducing the ollowing simpliications (Bianco 2008, Bianco et al. 2011): 1) a bi-linear rigid-sotening local bond stress-slip diagram was adopted instead 1 Post Doctoral Reseracher, Dept. o Structural Engrg. and Geotechnics, Sapienza University o Rome, via A. Gramsci 53, Rome, taly. vincenzo.bianco@uniroma1.it, corresponding Author. Tel , Fax Full Proessor, Dept. o Structural Engrg. and Geotechnics, Sapienza University o Rome, via A. Gramsci 53, Rome, taly. giorgio.monti@uniroma1.it. 3 Full Proessor, Dept. o Civil Engineering, University o Minho, Campus de Azurém, Guimarães, Portugal. barros@civil.uminho.pt. 1

2 o a multilinear diagram, 2) concrete racture surace was assumed as semi-pyramidal instead o semi-conical, 3) attention was ocused on the average-available-bond-length NSM FRP strip glued on the relevant prism o surrounding concrete, 4) determining the constitutive law o the average-available-bond-length NSM strip, along the approach ollowed or Externally Bonded Reinorcement (EBR) by Monti et al. (2003), and 5) determining the maximum eective capacity attainable by the average-available-bond-length NSM strip bridging the shear ailure crack, imposing a coherent kinematic mechanism (e.g. Monti et al. 2004, Monti and Liotta 2007). Nevertheless, even the second version o that model resulted not easy enough to be accepted by structural engineers since, in order to obtain the constitutive law o the average-available-bond-length NSM strip, it is necessary to implement an iterative procedure simulating the increasing value o the imposed end slip. n the present paper, the main physical eatures o a more simpliied version o that mechanical model are presented along with the urther simpliications introduced with respect to the ormer version. During the loading process o a RC beam subject to shear, when concrete average tensile strength ctm is attained at the web intrados (Fig. 1), some shear cracks originate therein and successively progress towards the web extrados. The governing shear ailure crack, herein designated as Critical Diagonal Crack (CDC), is inclined o an angle θ with respect to the beam longitudinal axis (Fig. 1a). The CDC can be schematized as an inclined plane dividing the web into two portions sewn together by the crossing strips (Fig. 1a). At load step t 1, the two web parts, separated by the CDC, start moving apart by pivoting around the crack tip whose trace, on the web ace, is point E in Fig. 1a. From that step on, by increasing the applied load, the CDC opening angle γ ( t n ) progressively widens (Fig. 1a). The strips crossing the CDC oppose its widening by anchoring to the surrounding concrete to which they transer, by bond, the orce originating at their intersection with the CDC, l O i, as a result o the imposed end slip δ [ γ ( t )], Fig. 1d. The capacity o each strip is provided by its available bond length L i that is the shorter Li n between the two parts into which the crack divides its actual length L (Fig. 1a). Bond is the mechanism through which stresses are transerred to the surrounding concrete (Yuan et al. 2004, Mohaed Ali et al and 2007, Bianco et al. 2009b). The local bond stress-slip relationship τ ( δ ), comprehensively simulating the mechanical phenomena occurring at 1) the strip-adhesive interace, 2) within the adhesive layer and at 3) the adhesive-concrete interace, can be represented, in a simpliied way, by a bi-linear curve (Fig. 1b). The subsuent phases undergone by bond during the loading process, representing the physical phenomena occurring in suence within the adhesive layer by increasing the imposed end slip, are: rigid, sotening riction and ree slipping (Fig. 1b) (Bianco 2008). The irst rigid branch (0-τ 0 ) represents the overall initial shear strength o the joint, independent 2

3 o the deormability o the adhesive layer and attributable to the micro-mechanical and mainly chemical properties o the involved materials and relative interaces. n act, the parameter τ 0 is the average o the ollowing physical entities encountered in suence by stresses lowing rom the strip to the surrounding concrete, i.e.: adhesion at the strip-adhesive interace, cohesion within the adhesive itsel, and adhesion at the adhesive-concrete interace (e.g. Sekulic and Curnier 2006, Zhai et al. 2008). The τ ( δ ) curve adopted (Fig. 1b) envisages that, by imposing an increasing end slip to the FRP strip, cracks orm instantaneously within the adhesive layer, both orthogonally to the (inclined) tension isostatics and along the strip-adhesive and adhesive-concrete interaces (e.g. Sena-Cruz and Barros 2004). Stresses are transerred by riction and micro-mechanical interlock along those micro-cracks. Nonetheless, by imposing an increasing end slip, those cracks progressively become smoother (sotening riction phase) up to the point ( δ Li = δ1 ) in which riction can no longer be mobilized and the strip is pulled out without having to overcome any restraint let (ree slipping phase). The constitutive law i ( Ri ; Li ) L δ o an NSM FRP strip, i.e. the orce transmissible by a strip with resisting bond length L Ri as unction o the imposed end slip δ Li, can be determined by analyzing the behavior o the simple structural element composed o the NSM FRP strip within a concrete prism (Fig. 1a,c-d) whose transversal dimensions are limited by the spacing s between adjacent strips and hal o the web cross section width b 2. w n this way, the problem o interaction between adjacent strips (Dias and Barros 2008, Rizzo and De Lorenzis 2009) is taken into account in a simpliied way, i.e., by limiting the concrete volume into which subsuent ractures can orm, to the amount o surrounding concrete pertaining to the single strip in dependence o s and b w. Moreover, even though the interaction with existing stirrups is herein neglected, it may be also accounted or by limiting the transversal dimension o the concrete prism to a certain ratio o b w 2, since the larger the amount o stirrups, the shallower concrete racture is ected to be, even i urther research is necessary in this respect. Attention can be ocused on the system composed o the strip with the average value o available bond length glued on the pertaining prism o surrounding concrete (Fig. 1c-d). The ailure modes o an NSM FRP strip subject to an imposed end slip comprise, depending on the relative mechanical and geometrical properties o the materials involved: debonding, tensile rupture o the strip, concrete semi-pyramidal tensile racture, and a mixed shallowsemi-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-concrete interaces, so that the strip pulling out results (Fig. 1e). When principal tensile stresses transerred to the 3

4 surrounding concrete attain its tensile strength, concrete ractures along a surace, envelope o the compression isostatics, whose shape can be conveniently assumed as a semi-pyramid with principal generatrices inclined o an angle α with respect to the strip longitudinal axis (Fig. 1c-d). ncreasing the imposed end slip can result in subsuent semi-pyramidal and coaxial racture suraces in the concrete surrounding the NSM strip. This concrete racture process progressively reduces the resisting bond length length L Ri that is the portion o the initial available bond L i still bonded to concrete. Those subsuent ractures can either progress up to the ree end, resulting in a concrete 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 concrete racture, the strip can rupture (Fig. 1e). This modeling strategy can be urther simpliied introducing the ollowing assumptions: 1) concrete racture can be accounted or determining the uivalent value o the average resisting bond length L Ri, which is the portion o the available average resisting bond length L Ri = η L ( 0 η 1 ) that would still be adhered to surrounding Ri concrete, as unction o the concrete mechanical properties, ater all the successive co-axial semi-pyramidal racture suraces have ormed in the surrounding concrete, and 2) the post peak behavior o the bond-based bd constitutive law i ( Ri; Li) L δ o the uivalent value o the average resisting bond length can be neglected. As to the irst assumption, the value o concrete average tensile strength * or values larger than which concrete does ctm not racture at all ( η = 1) can be determined by imposing the uality between the maximum value o orce that L Ri can transer through bond stresses, and the corresponding value o concrete semi-pyramidal racture strength. n other words, instead o imposing an increasing value o imposed end slip δ Li to L Ri and analyzing how it progressively reduces at the occurrence o successive co-axial concrete ractures (Bianco et al. 2011), one can assume that, i concrete does not racture or the value o δ Li in correspondence o which the orce transerred through bond by L has attained its maximum value, even more it will not racture either or smaller values o Ri δ Li, or or larger values o δ Li. This is due to the act that the maximum value o orce that L Ri can transer through bond is attained or the value o δ Li in correspondence o which the distribution o bond stresses has reached the strip ree extremity, i LRi L Re, and when the bond transer length has matched the eective resisting bond length, i LRi > LRe. For a null value o, the assumption is made that concrete semi-pyramidal ctm 4

5 successive co-axial racture suraces would reach the average available bond length ree extremity resulting in Ri * L = 0 ( η = 0 ). t is also assumed that, or values o concrete tensile strength < <, the reduction actor η has a linear trend between zero and unit 0 η 1 (Fig. 2). The maximum value o the eective capacity 0 ctm ctm max i, e, which is the maximum value o the average o the NSM FRP strip capacity along the CDC (Bianco et al. 2011), can be approximated, and lightly underestimated, neglecting the post peak behavior o the bond-based constitutive law o the uivalent resisting bond length. n the ollowing paragraphs the resulting analytical uations are irstly presented and then are applied to simulate some o the most recent erimental results. Moreover, Monte Carlo simulation are carried out in order to assess the sensitivity o the results provided by the proposed ormulation to the value assumed by the input parameters. Proposed design ormula The input parameters include (Fig. 1): beam cross-section web s depth h w and width b w ; inclination angle o both CDC and strips with respect to the beam longitudinal axis, θ and β, respectively; strips spacing measured along the beam axis s ; angle α between axis and principal generatrices o the semi-pyramidal racture surace (Fig. 1c-d); concrete average compressive strength cm ; FRP tensile strength u and Young s modulus E ; thickness a and width b o the strip cross-section, and values o bond stress τ 0 and slip δ 1 deining the adopted local bond stress-slip relationship (Fig. 1b). The implementation o the proposed calculation procedure comprehends the ollowing steps (Fig. 3): 1) evaluation o the average value o the available (resisting) bond length L and o the minimum integer number N,int o Ri l NSM strips that can eectively cross the CDC; 2) evaluation o various constants, both integration and geometric ones; 3) evaluation o the reduction actor η o the average value o the available (resisting) bond length and o the uivalent value o the average resisting bond length L Ri ; 4) evaluation o the value o the imposed end slip δ Lu in correspondence o which the peak value o the orce transmissible through bond by the uivalent value o the resisting bond length L Ri can be attained; 5) evaluation o the maximum eective capacity that a NSM o bond length can attain during the beam loading process ( contribution L Ri. max i, e ), and 6) evaluation o the strips shear strength 5

6 Average value o the available resisting bond length L and minimum number o strips N,int eectively Ri l crossing the CDC The average value o the available bond length and the minimum integer number o strips eectively crossing the CDC can be evaluated as ollows (Fig. 1): N L l,int Ri hw sinθ ( cotθ + cot β ) = 4 sin ( θ + β ) = round o hw ( cotθ + cot β ) s (1) (2) Evaluation o Constants t is necessary to calculate some constants, both geometrical and integration ones, characterizing the bond transer mechanism o the adopted NSM system, composed o a certain type o FRP strips, adhesive and concrete (Bianco et al. 2011). The geometric constants encompass (Fig. 1): L = 2 b + a (3) p the eective perimeter o the strip cross section; bw Ac = s (4) 2 the cross section area o the relevant prism o surrounding concrete; and the CDC length. The mechanical constants encompass: L d hw = (5) sinθ tr u = a b (6) the strip tensile strength; concrete mean tensile strength; and ctm concrete s Young s modulus, where both the CEB-FP Model Code 1990, with cm in MPa. The bond-modeling constants encompass (see the Appendix): c = 1.4 ( ( 8) 10) 2 3 (7) cm E = ( 10) 1 3 (8) cm E c and ctm are herein evaluated by means o the ormulae present in 6

7 Lp 1 A J1 = + A E Ac Ec 1 δ 1 ; 2 λ = τ0 J1 ; tr J1 C3 = L λ p (9) that are integration constants regarding the bond transer mechanism; L Re π = ; 2 λ bd 1 Lp λ δ1 = (10) J 1 the eective resisting bond length L, and the corresponding maximum bond orce 1. Re bd Reduction actor η and uivalent value o the average resisting bond length The reduction actor can be evaluated as ollows: L Ri η ( s ; bw; cm; LRi ) * * ctm ctm ctm ctm se < = 1 se ctm * ctm (11) where: * ctm = ( λ LRi ) ( ) Lp λ δ1 sin bw J1 min LRi tan α; min s sin β; 2 LRi tanα 2 (12) in which L Ri has to be set ual to: L Ri L i L L = L i L > L Ri Ri Re Re Ri Re (13) The value o * is the one corresponding, under the above simpliying assumptions, or a given average NSM ctm system, to the limit condition o occurrence o concrete racture. The uivalent value o the average resisting bond length is given by: ( ; ; ; ) Ri Ri η w cm Ri L = L s b L (14) alue o the imposed end slip δ Lu in correspondence o which the peak value o the comprehensive constitutive law ( i Ri; Li) L δ o the uivalent average resisting bond length L Ri is attained n case the maximum invariant value o orce that a given NSM system, characterized by a given FRP strip, a given adhesive layer and concrete, can transer through bond stresses db 1 is larger or ual to the strip tensile strength tr, the value o imposed end slip in correspondence o which the peak value o the constitutive law i ( LRi; δ Li) tr is assumed as the minimum between δ L1 ( LRi ) and δ L1 ( ) (Fig. 2d), where L1 ( LRi ) δ is the value o imposed 7

8 bd end slip in correspondence o which the bond-based constitutive law i ( Ri; Li) tr 1( ) L L δ attains the peak value, and δ is the value o the imposed end slip in which the bond transerred orce uals the strip tensile strength tr u ( = a b ). The value o the imposed end slip δ Lu in correspondence o which the peak orce transmissible by the uivalent average resisting bond length L Ri is attained, is given by: δ ( ) δ ( ) δ ( ) bd tr L1 LRi se 1 δ < = Lu tr bd tr min L1 LRi ; Li se 1 (15) where: δ L1 ( LRi ) ( λ ) Ri Ri Re δ1 1 cos L or L L = δ1 or LRi > L Re (16) and: δ Li C δ tr 3 ( ) = δ1 1 cos arcsin 1 (17) max i, e Maximum eective capacity o the NSM strip with uivalent average resisting bond length The maximum eective capacity can be evaluated, neglecting the post peak behavior o the constitutive law, as ollows (Bianco et al. 2011): L Ri δ1 A2 π = i e ( γ ) = arcsinψ ψ 1 ψ 2 L A γ 2 max 2 i, e, max d 3 max (18) where: L p λ sin( θ + β ) A2 = ; A3 = ; γ J 2 δ 1 1 max = L d 2 δlu sin ( θ + β ) ; ψ 1 A3 γ max Ld = (19) Shear strength contribution provided by a system o NSM FRP strips The actual and design value d o the NSM shear strength contribution, can be obtained as ollows: 1 1 l ( max d = 2 N,int i, e sin β ) γ = γ (20) Rd Rd 8

9 where γ Rd is the partial saety actor, divisor o a capacity, that can be assumed as according to the level o uncertainty aecting the input parameters but, in this respect, a reliability-based calibration is needed. Formulation Appraisal The proposed ormulation was applied to the RC beams tested by Dias and Barros (2008), Dias et al. (2007), Dias (2008), Rizzo and De Lorenzis (2009), A.K.M. Anwarul slam (2009), Rahal, K.N. (2010), Rahal and Rumaih (2011), Jalali et al. (2012) and Cisneros et al. (2012). The beams tested in the irst two erimental programs (series and ) were T cross-section RC beams characterized by the same test set-up with the same ratio between the shear span and the beam eective depth ( a d = 2.5 ), the same amount o longitudinal reinorcement, the same kind o CFRP strips and epoxy adhesive and they diered or the concrete mechanical properties. n act, the irst erimental program (series ) was characterized by a concrete average compressive strength cm o 31.1 MPa, while the second (series ) cm =18.6 MPa. Both series presented dierent conigurations o NSM strips, in terms o both inclination β and spacing s. The second program also included beams characterized by a dierent amount o existing steel stirrups (see Table 1). The beams tested in the third erimental program (series ) were characterized by the same test set up, but with a dierent shear aspect ratio ( a d = 3.3 ) and a higher concrete average compressive strength ( cm = 59.4 MPa ). Some o them were also subjected to pre-cracking (their label includes a letter F). Those beams are characterized by the ollowing coon geometrical and mechanical parameters: b = 180 ; hw = 300 ; u = 2952 MPa (or the series and ) and u = 2848 MPa (or the series ); E = GPa (or the series and ) and E = GPa (or the series ); a = 1.4 ; b = The CDC inclination angle θ adopted in the simulations by the proposed ormulation, listed in Table 1, is the one erimentally observed by inspecting the crack patterns (Dias 2008). Note that the erimental observations conirm the ected trend according to which θ diminishes or increasing values o the ratio a d (e.g. w Bousselham and Chaalal 2004, Chao et. al. 2005). n act, or some beams o the series ( a d = 3.3 ), θ assumes values smaller than 45º and up to 20º (Table 1). n this respect, it has to be stressed that assuming θ = 45º could result excessively conservative since, with respect to smaller values (e.g. θ = 20º ), and other parameters being the same, the predicted NSM shear strength contribution decreases due to the act that the number o strips eectively crossing the CDC diminishes (Bianco 2008). t would be necessary to develop rigorous uations to 9

10 evaluate the CDC inclination angle θ as unction o 1) shear aspect ratio a d and amount o both 2) NSM strips and 3) existing steel stirrups but, in this respect, urther research is necessary. The beams tested by Rizzo and De Lorenzis (2009) (series ) were rectangular cross-section RC beams strengthened in shear by either bars (their label starts by NB) or strips (their label starts by NS) and tested under our point bending (Table 2). These beams were characterized by cross-section dimensions o b = 200 and w hw = 210, and the ratio between the shear span and the beam eective depth was a d = 3.0. Concrete had average compressive strength o 29.3 MPa. The round CFRP bars have a 8 diameter cross-section and or improvement o the bond properties, the surace o this type o bar is spirally wound with a carbon iber tow and sand coated. The tensile strength and modulus o elasticity o the bars were = 2.21 GPa and E = GPa, respectively. The strips, have a cross-section o dimensions a = 2.0 and b = 16.0, and mechanical properties o = 2.07 GPa and E = GPa. Two kinds o epoxy were employed to glue the NSM u reinorcement, both two-component 100% solid non-sag tixotropic epoxy adhesive pastes obtained by mixing resin and hardener in a 3:1 weight ratio, dier by the values o tensile strength and modulus o elasticity and are indicated, in the beam codes, by a letter a or b. The tensile strength and the secant tensile elastic modulus presented values o 18.6 MPa and 4.15 GPa, respectively, or the type-a adhesive, and values o 22.8 MPa and GPa or the type-b. Since the erimentally observed value o the CDC inclination angle is not reported, in the simulation a value o 45 was assumed. The beams tested by A.K.M. Anwarul slam (2009) (series ) were rectangular cross-section RC beams strengthened in shear by CFRP round bars and tested under our point bending (Table 3). These beams were characterized by cross-section dimensions o b = 254 and h = 305, and the ratio between the shear w span and the beam eective depth was a d = Concrete had average compressive strength o MPa. The round CFRP bars have a 9 diameter cross-section and tensile strength and modulus o elasticity o the bars were = 2.07 GPa and E = GPa, respectively. CFRP bars are inclined o 90 with respect to the beam u longitudinal axis and since the erimentally observed value o the CDC inclination angle is not reported, in the simulation a value o 45 was assumed. The beams tested by Rahal (2010) and by Rahal and Rumaih (2011) (series ) were T cross-section RC beams strengthened in shear by NSM bars either o steel (their label includes a letter R) or CFRP (their label includes a letter F) and tested under our point bending (Table 4). All the ive tested beams contained two test regions each, one strengthened with CFRP bars and the other with conventional steel bars. Hence, a total o ten results are w u 10

11 reported. These beams were characterized by T cross-section dimensions o b = 150 and h = 400, while the lange was 380 in width and 100 in depth, and the ratio between the shear span and the beam eective depth was a d = 3.0. Concrete average compressive strength ranged rom 36.2 MPa to 38.6 MPa. The epoxy resin used to grout the bars in the groves was non-sag resin. The NSM steel bars were characterized by yield tensile strength o 0.51 GPa, while the NSM FRP bars were 8 deormed TYFO carbon FRP bars characterized by = 1.90 GPa and E = GPa. Since the erimentally observed value o the CDC u inclination angle is not reported, in the simulation a value o 45 was assumed. The beams tested by Jalali et al. (2012) (series ) were rectangular cross-section RC beams strengthened in shear by Manually Made NSM FRP rods (MMFRPs) with (their label includes a letter A) and without an extremity anchorage (Table 5) and tested under three point bending. These particular kind o FRP rods were obtained by wrapping a dry carbon iber sheet, which was pre impregnated with resin, around a 6 diameter wooden rod. u These rods were characterized by = 3.55 GPa and E = GPa. The beams were rectangular crosssection with dimensions b = 200 and h = 250, and the ratio between the shear span and the beam w w eective depth was a d = Concrete average compressive strength was ual to = 36.4 MPa. Since the erimentally observed value o the CDC inclination angle is not reported, in the simulation a value o 45 was assumed. The beams tested by Cisneros et al. (2012) (series ) were rectangular cross-section RC beams strengthened in shear by either bars (their label starts by B) or strips (their label starts by S) and tested under three point bending (Table 6). Beams cross-section dimensions were b = 200 and h = 350. Each beam was tested twice, w once at each end, and the ratio between the shear span and the beam eective depth ual to a d = 2.9. Concrete average compressive strength ranged rom = MPa to = MPa. The NSM FRP bars were cm characterized by 8 diameter, while the strips had cross section dimensions o a = 2.5 and b = FRP mechanical properties were = 2.5 GPa and E = GPa. The resin used was MBrace Adhesive u 220 or the bars and Masterlow 920 SF or the strips. Since the erimentally observed value o the CDC inclination angle is not reported, in the simulation a value o 45 was assumed. The angle α was assumed ual to 28.5 or all the erimental programs, being the average o values obtained in a previous investigation (Bianco 2008) by back-analysis o erimental data. As to the value o α, due to its importance to the prediction accuracy o NSM shear strength contributions, urther research is desirable. Concrete cm w w cm w 11

12 average tensile strength ctm was calculated rom the average compressive strength by means o the ormulae o the CEB Fib Model Code The parameters characterizing the adopted local bond stress-slip relationship (Fig. 1b) are or all the erimental programs: τ 0 = 20.1 MPa and δ 1 = Those values were obtained by the values characterizing the more sophisticated local bond stress-slip relationship adopted in previous works (Bianco et al. 2009a, 2010), by ixing the value o τ 0 = 20.1 MPa and determining δ 1 = 7.12 by uating the racture energy. n this respect, it has to be underlined that the necessity is elt to develop rigorous uations that would allow the values ( τ, δ ) characterizing the local bond stress slip relationship to be determined on the basis 0 1 o: a) supericial chemical and micro-mechanical properties o FRP, adhesive and concrete, and b) the adhesive layer thickness. Nonetheless, urther research is, in this respect, ruired. However, as highlighted by means o parametric studies (Bianco 2008), or the values o concrete mechanical properties that can be met in practice, debonding rarely occurs due the high capacity o currently available structural adhesives. Thus, slight variations o the values o the parameters τ 0 and δ 1 cannot be elt, in terms o NSM shear strength contribution, due to the premature occurrence o other ailure modes such as either concrete racture or strip rupture. For this reason, adopting values o τ 0 = 20.1 MPa and δ 1 = 7.12 or cases characterized by dierent values o both 1) supericial chemical-mechanical properties o FRP, adhesive and concrete, and 2) adhesive thickness, is not ected to signiicantly aect the predictive perormance o the proposed ormulation. When NSM round bars are employed instead o strips, the uivalent square cross-section is employed in the calculations. n the case o the MMFRPs adopted by Jalali et al. (2012), since the FRP cross-section is an annulus, the corresponding area is adopted to evaluate A (with A 2 = obtaining a = b = 7.10 ). The same values o the uivalent square cross section dimensions are adopted to evaluate the eective perimeter L = 2 b + a, even though slightly underestimating this latter. p Note also that, or the works among those adopted as basis o the model appraisal, not reporting directly the NSM shear strength contribution, this latter has been herein obtained as the dierence between the shear strength o the strengthened beam and that o the control beam. Note also that in Tables 1-6, the symbol h w has been adopted to indicate also the vertical height o the NSM bars, when they are coincident while, when the NSM bar is inserted through the lange thickness, the symbol h has been adopted instead. The predictions obtained by the ormulation proposed in the present work have also been compared with those obtained by the previous two more sophisticated versions o the same modeling strategy. The predictions obtained 12

13 by the irst model (Bianco et al. 2009a), which contemplates three geometrical conigurations that the occurred CDC can assume with respect to the system o NSM FRPs, are labeled by,1,,2 and,3. The predictions obtained by the second and simpliied model (Bianco et al. 2011) are labeled by third, urther simpliied model herein presented are labeled by while those obtained by the. The erimental results concerning beams 2S-8L45- and 4S-7L- have been neglected in the considerations below since they are deemed aected by some disturbance. The ormulation herein proposed provides, in general, satisactory estimates o the erimental recordings since the ratio presents altogether mean value and standard deviation ual to 0.69 and 0.29, respectively. The second version o the adopted modeling strategy also provides satisactory estimates o the erimental recordings since the ratio presents altogether mean value and standard deviation ual to 0.67 and 0.29, respectively, even it is not closed orm and ruires the implementation o an iterative procedure to determine the average resisting bond length strip s constitutive law. The irst version o the adopted modeling strategy, much more sophisticated than the two more recent versions, provides much more satisactory estimates. n act, considering, or each analyzed beam, the closest prediction out o the three obtained, the ratio presents altogether mean value and standard deviation ual to 0.98 and 0.26, respectively, even i it is very cumbersome to be implemented and needs a stand-alone sotware to be developed. Sensitivity analysis A sensitivity analysis was carried out to assess the relative importance o each input parameter on the calculated value o the NSM shear strength contribution in order to igure out what are the input parameters that mostly aect the result. For this purpose, the proposed ormulation, presented in the previous sections, was implemented in a spreadsheet that was re-calculated iteratively one hundred thousand times, each time with a set o new possible values o the input parameters. At each iteration, the new value o each parameter was sampled, independently rom each other, rom the relevant probability distribution assigned to it, and a new value o the output was generated in the corresponding cell. As simulation progressed, new possible outcomes were generated rom each iteration and a numerical solver kept track o these output values. All o the input parameters ( a, b, s, b w, h w, cm, α, u, E, θ, τ 0, δ 1 ) were characterized by a uniorm probability distribution, which means a range 13

14 o possible values with the same likelihood o occurrence (Table 7). These intervals cover the possible situations occurring in NSM shear strengthening interventions or RC beams. n particular: the strip thickness a was characterized by a range o variation between 1. 0 and 5.0 ; the strip width b, the strips spacing s, and the concrete average compressive strength cm were characterized by ranges o variation 5 35, and MPa, respectively; beam cross-section web s depth h w and width b w were characterized by ranges o variation and The input parameters τ 0 and δ 1 were characterized by ranges o variation MPa and , respectively. At each iteration, the value o each input parameter was sampled rom the relevant probability distribution by the Monte Carlo Sampling Technique (Law and Kelton 2000, Ang and Tang 1975). Monte Carlo simulations is simply a repeated process o generating deterministic solutions to a given problem; each solution corresponds to a set o deterministic values o the underlying random input parameters. n other words, a Monte Carlo simulation recalculates the worksheet in which the deterministic ormulation was implemented, over and over, each time assuming a set o input parameters whose values are selected randomly rom the probability distribution assigned to each o them. The outcome o the above simulations, being the result o calculations in which all input parameters are characterized by a uniorm probability distribution, is itsel characterized by a probability distribution and can be represented both in terms o probability density and cumulated ascending probability (Fig. 4). From those results it arises that, or the range o variations herein assigned to the input parameters (Table 7): 1) the uivalent average max resisting bond length strip s maximum eective capacity i, e ( Ri) L varies between 0.0 and 40.8 in 95 % o cases, and 2) the NSM FRP strips shear strength contribution varies between 0.0 and in 95 % o cases. The sensitivity analysis between the output and the input variables was herein carried out evaluating, by the multivariate stepwise regression method (e.g. Draper and Smith 1966), the regression coeicients o the input variables (Fig. 4). The larger the coeicient, the larger the impact that particular input has on the calculated value. A positive coeicient, with bar extending to the right, indicates that this input has a positive impact, which means that increasing this input will increase the output, while the opposite happens i the coeicient is negative. The values o the plotted regression coeicients (Fig. 4) indicate the increment o output or a standard deviation o the relevant input. For instance, h w has a mapped coeicient o 86.50, meaning that an increase o µ raction o a standard deviation o input parameter h w yields an increase o µ units (, not standard deviations) 14

15 o the output. The standard deviation o sampled values o h w is (Table 8) and, thereore, every unit increase o input h w impacts the output positively by = From the sensitivity analysis it was ound that the input parameters that mostly aect the value o are, in decreasing order o impact, the angle α between axis and generatrices o the concrete racture surace, the concrete max i, e average compressive strength cm, the beam cross section depth h w, and the strips spacing s (Fig. 4). t was also observed that the value o θ (Fig. 4). is mostly aected by the value o h w, α, cm, and the CDC inclination angle Conclusions A design ormulation to predict the NSM FRP strips shear strength contribution to a RC beam was derived rom a previously developed numerical model, introducing some simpliications, such as: 1) assuming the phenomenon o concrete racture just as a reducer o the average available resisting bond length, and evaluating a resulting uivalent average resisting bond length, and 2) neglecting the post peak behavior o the uivalent average resisting bond length s bond-based constitutive law in the evaluation o the strip s maximum eective capacity. The predictive perormance o the ormulation was appraised by considering some o the most recent erimental results available in literature. The ormulation provided very satisactory estimates o the erimental recordings, resulting the ratio o the prediction versus the erimental value characterized by a mean value and a standard deviation o 0.69 and 0.29, respectively. The proposed ormulation was subsuently employed to carry out Monte Carlo simulations sampling, at each iteration, the value o each input parameter rom the relevant uniorm probability distribution and independently rom each other. The results o those simulations were adopted in order to igure out which input parameters mostly aect the prediction o the NSM FRP strips shear strength contribution to a RC beam. From this sensitivity it arises that the input parameters that mostly aect the NSM shear strength contribution prediction are, in decreasing order o importance: h w, α, cm and the CDC inclination angle θ. Acknowledgements 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 PrePam, with reerence number o PTDC/ECM/114511/2009, supported by FCT. Also, this work was 15

16 carried out under the auspices o the talian DPC-ReLuis Project (repertory n. 540), Research Line 8, whose inancial support is greatly appreciated. 16

17 Notation A c = area o the concrete prism cross section A = area o the strip s cross section A 2 = integration constant entering the ressions to evaluate the A 3 = integration constant entering the ressions to evaluate the C 3 = integration constant or the sotening riction phase E c = concrete Young s modulus E = strips CFRP Young s modulus J 1 = bond modeling constant L d = CDC length L Ri = average available resisting bond length max i, e max i, e L p = eective perimeter o the strip cross section L Ri = i-th strip resisting bond length L Re = Eective resisting bond length L Ri = Equivalent average resisting bond length l,int N = minimum integer number o strips that can eectively cross the CDC OXYZ = crack plane reerence system O = reerence axis along the i-th strip available bond length L i l l i X i c i = progressive concrete tensile racture capacity along the i-th strip db 1 = Maximum value o orce transerable through bond by the given FRP NSM system max i, e = maximum eective capacity = erimental value o the NSM shear strengthening contribution tr = Strip tensile rupture capacity = actual value o the NSM shear strengthening contribution 17

18 d = design value o the NSM shear strengthening contribution ( ; ) L δ = comprehensive constitutive law o the average available resisting bond length i Ri Li ( ; ) bd i Ri Li L δ = bond-based constitutive law o the average available resisting bond length a = strip cross section s thickness b = strip cross section s width b w = beam cross section s width cm = concrete average cylindrical compressive strength ctm = concrete average tensile strength * ctm = alue o concrete average tensile strength or values larger than which concrete racture does not occur u = FRP tensile strength h w = beam web height h = vertical height o the NSM bar s = spacing between adjacent strips along the beam axis α = angle deining the concrete racture surace β = FRP strips inclination angle with respect to the beam longitudinal axis δ 1 = slip corresponding to the end o sotening riction δ Li = imposed slip at the loaded extremity o the i-th strip δ Lu = imposed slip in correspondence o which the comprehensive peak orce transmissible by L Ri is attained tr 1( ) δ = L 1 ( LRi ) L alue o imposed end slip in correspondence o which the strip tensile strength is attained δ = value o δ Li deining the end o the irst phase o the bond-based constitutive law γ max = CDC opening angle or which the maximum eective capacity is attained η = reduction actor o the initial average available resisting bond length γ Rd = partial saety actor divisor o the capacity λ = constant entering the governing dierential uation or elastic phase 18

19 θ = critical Diagonal Crack (CDC) inclination angle θ = erimentally observed CDC inclination angle τ ( δ ) = local bond stress-slip relationship ψ = Constant necessary to evaluate the maximum eective capacity provided by the uivalent average resisting bond length τ 0 = adhesive-cohesive initial bond strength 19

20 Reerences Ang, A., H-S, Tang, (1975), Probability Concepts in Engineering Planning and Design olume Basic Principles, John Wiley & Sons, New York. Ang, A., H-S, Tang, (1975), Probability Concepts in Engineering Planning and Design olume Decision, Risk and Reliability, John Wiley & Sons, New York. Bianco,., Barros, J.A.O., Monti, G., (2009a). Three dimensional mechanical model or simulating the NSM FRP strips shear strength contribution to RC beams, Engineering Structures, 31(4), April 2009, Bianco,., Barros, J.A.O., Monti, G., (2009b). Bond Model o NSM FRP strips in the context o the Shear Strengthening o RC beams, ASCE Journal o Structural Engineering, 135(6), June Bianco,., Barros, J.A.O., Monti, G., (2010). New approach or modeling the contribution o NSM FRP strips or shear strengthening o RC beams, ASCE Journal o Composites or Construction, 14(1), January/February Bianco,., (2008). Shear Strengthening o RC beams by means o NSM FRP strips: erimental evidence and analytical modeling, PhD Thesis, Dept. o Structural Engrg. and Geotechnics, Sapienza University o Rome, taly, submitted on December Bianco,., Monti, G., Barros, J.A.O., (2011). Theoretical model and computational procedure to evaluate the NSM FRP strips shear strength contribution to a RC beam, ASCE Journal o Structural Engineering, 137(11), November Bousselham A., Chaalal O., (2004) Shear Strengthening Reinorced Concrete Beams with Fiber-Reinorced Polymer: Assessment o nluencing Parameters and Ruired Research, AC Structural Journal, ol.101, Nº2, March-April, pp CEB-FP Model Code 90, (1993) Bulletin d normation N 213/214, Final version printed by Th. Telord, London, (1993; SBN ; 460 pages). Chao S.Y., Chen J.F., Teng J.G., Hao Z., Chen J., (2005) Debonding in Reinorced Concrete Beams Shear Strengthened with Complete Fiber Reinorced Polymer Wraps, Journal o Composites or Construction, ASCE September/October 2005/1. Cisneros, D., Arteaga, A., De Diego, A., Alzate, A., Perera, R., (2012), Experimental study on NSM FRP shear retroitting o RC beams, 6 th international conerence on Composites in Civil Engineering, CCE 2012, Rome, taly, June

21 Dias, S.J.E. (2008). Experimental and anlytical research in the shear strengthening o reinorced concreet beams using the near surace mounted technique with CFRP strips, PhD Thesis, Department o Civil Engineering, University o Minho, Guimarães-Portugal, in Portuguese. Dias, S.J.E., Bianco,., Barros, J.A.O., Monti, G., (2007). Low strength concrete T cross section RC beams strengthened in shear by NSM technique, Workshop-Materiali ed Approcci nnovativi per il Progetto in Zona Sismica e la Mitigazione della ulnerabilità delle Strutture, University o Salerno, taly, February. Dias, S.J.E. and Barros, J.A.O., (2008). Shear Strengthening o T Cross Section Reinorced Concrete Beams by Near Surace Mounted Technique, Journal o Composites or Construction, ASCE, ol. 12, No. 3, pp Draper, N.R., and Smith, H., (1966). Applied Regression Analysis, New York: John Wiley & Sons, nc. Chapter 6. slam, Anwarul, A.K.M., (2009). Eective methods o using CFRP bars in shear strengthening o concrete girders, Engineering Structures, 31(2009), Jalali, M., Sharbatdar, M.K., Chen, Jian-Fei, Alaee, Farshid Jandaghi, (2012), Shear strengthening o RC beams using innovative manually made NSM FRP bars, Construction and Building Materials, 36, Law, A.M. and Kelton, W.D. Simulation Modeling and Analysis: McGraw-Hill, New York, NY, 1991,1982, Monti, G., Renzelli, M., Luciani, P., (2003) FRP Adhesion to Uncracked and Cracked Concrete Zones, Proceedings o the 6th nternational Symposium on Fibre-Reinorced Polymer (FRP) Reinorcement or Concrete Structures (FRPRCS-6), Singapore, July, Monti, G., Santinelli, F., Liotta, M.A., (2004) Mechanics o FRP Shear Strengthening o RC beams, Proc. ECCM 11, Rhodes, Greece. Monti, G., Liotta, M.A., (2007) Tests and design uations or FRP-strengthening in shear, Construction and Building Materials (2006), 21(4), April 2007, Mohaed Ali, M.S., Oehlers, D.J., Seracino, R. (2006). ertical shear interaction model between external FRP transverse plates and internal stirrups, Engineering Structures 28, Mohaed Ali, M.S., Oehlers, D.J., Griith, M.C., Seracino, R. (2007). nteracial stress transer o near surace-mounted FRP-to-concrete joints, Engineering Structures 30, Rahal, K.N., (2010). "Near surace mounted shear strengthening o reinorced concrete beams," American Concrete nstitute, Special Publication SP , Oct. 2010, pp

22 Rahal, Khaldoun N., Rumaih, Hantem A., (2011), Tests on reinorced concrete beams strengthened in shear using near surace mounted CFRP and steel bars, Engineering Structures, 33, Rizzo, A. and De Lorenzis, L., (2009) Behaviour and capacity o Rc beams strengthened in shear with NSM FRP reinorcement, Construction and Building Materials, ol. 3, n. 4, April 2009, Sekulic, A., Curnier, A., (2006). An original epoxy-stamp on glass-disc specimen exhibiting stable debonding or identiying adhesive properties between glass and epoxy, nternational Journal o Adhesion and Adhesives, ol. 27, pp Sena-Cruz, J.M., Barros, J.A.O., (2004). Bond between near-surace mounted CFRP laminate strips and concrete in structural strengthening, Journal o Composites or Construction, ASCE, ol. 8, No. 6, pp Yuan, H., Teng, J.G., Seracino, R., Wu, Z.S., Yao, J. (2004). Full-range behavior o FRP-to-concrete bonded joints, Engineering Structures, 26, Zhai, L.L., Ling, G.P., Wang, Y.W., (2008). Eect o nano-al 2O 3 on adhesion strength o epoxy adhesive and steel, nternational Journal o Adhesion and Adhesives, ol. 28, No. 1-2, pp

23 TABLE CAPTONS Table 1. alues o the parameters characterizing the beams tested by Dias and Barros (2008), Dias et al. (2007) and Dias (2008). Table 2. alues o the parameters characterizing the beams tested by Rizzo and De Lorenzis (2009). Table 3. alues o the parameters characterizing the beams tested by A.K.M. Anwarul slam (2009). Table 4. alues o the parameters characterizing the beams tested by Rahal and Rumaih (2011) and by Rahal (2010). Table 5. alues o the parameters characterizing the beams tested by Jalali et al. (2012). Table 6. alues o the parameters characterizing the beams tested by Cisneros et al. (2012). Table 7. alues characterizing the uniorm probability distribution o the input parameters. Table 8. Detailed statistics o both the sampled values o input parameters and the calculated output. 23

24 Table 1. alues o the parameters characterizing the beams tested by Dias and Barros (2008), Dias et al. (2007) and Dias (2008). Beam Label θ β s Steel Stirrups,1,2,3 2S-3L F6/ S-5L S-8L S-3L S-5L S-8L * S-3L S-5L S-7L S-7L F6/ S-4L S-7L S-4L S-6L S-7L F6/ * S-4L S-7L S-4L S-6L S-5L F6/ S-5L45F1- ** S-5L45F2- ** S-5L F6/ S-5L45F- ** S-9L F6/ S-9L F6/ S-5L F6/ S-5L F6/ S-5L60F- ** S-8L F6/ S-8L F6/ S-6L F6/ S-10L * beams whose erimental value o NSM shear strength contribution is aected by some disturbance; ** beams that were subjected to pre-cracking. 24

25 Table 2. alues o the parameters characterizing the beams tested by Rizzo and De Lorenzis (2009). Beam Label θ β s u GPa E GPa a b φ,1,2,3 NB90-73-a NB90-73b NB90-45-b NB a NB45-73-a NS90-73-a NS a b w 200 ; h w 210 ; cm 29.3 MPa; steel stirrups F6/160; 25