ANALYTICAL AND EXPERIMENTAL STUDY OF FRP-STRENGTHENED RC BEAM-COLUMN JOINTS. Abstract

Transcription

1 ANALYTICAL AND EXPERIMENTAL STUDY OF FRP-STRENGTHENED RC BEAM-COLUMN JOINTS Dr. Costas P. Antonopouos, University of Patras, Greece Assoc. Prof. Thanasis C. Triantafiou, University of Patras, Greece Abstract Anaytica modes are presented in this study for the anaysis of RC joints strengthened with composite materias. The modes provide equations for stresses and strains at various stages of the response (before or after yieding of the beam or coumn reinforcement) unti the utimate capacity is reached, defined by concrete crushing or FRP faiure due to fracture or debonding. Soutions to these equations are obtained numericay. The anaytica formuation provides usefu information on the shear capacity of FRP-strengthened joints in terms of the quantity and configuration of the externay bonded reinforcement and may be used to design FRP jackets for poory detaied beamcoumn joints; an iustration is provided through a case study. Finay, the anaytica mode is compared with a series of test resuts and the agreement between theory and experiments is found satisfactory. Introduction To overcome the difficuties and some probems associated with traditiona techniques for strengthening shear-critica RC joints, namey intensive abor, artfu detaiing, increased dimensions, corrosion protection and specia attachments, recent research efforts have focused on the use of fiber reinforced poymers (FRP), that may be epoxy-bonded in the form of fexibe sheets or strips, with fibers oriented propery so as to carry tension forces due to shear (see Figure 1a as a typica exampe). coumn beam h t (a) d (b) Figure 1. (a) Schematic iustration of RC joint strengthened with FRP (ines indicate direction of fibers). (b) Moment and shear acting at joint and definition of coordinate system. Unti now, FRP-strengthened joints have been studied mainy experimentay. Anaytica modeing has been extremey imited and has ed to oversimpified design approaches, faiing to capture the rea state of stress (and strain) in the joint. 1

2 On the other hand, anaytica modeing of RC joints (without FRP) has been extensive. One of the most powerfu modes is that of Pantazopouou and Bonacci (199), which uses stress equiibrium and strain compatibiity to yied the shear strength of a joint with known geometry and reinforcement quantities. In this study the authors have extended the aforementioned mode to account for the effect of externay bonded FRP. They have aso deveoped computer programs that may be used to trace the state of stress and strain in RC joints strengthened with either unidirectiona strips or fexibe sheets (the atter may be combined to form aminates). Foowing the anaytica and numerica formuations one case study is anayzed and a comparison of the anaytica mode with existing test resuts is made. Joints Strengthened with Sheets Basic Assumptions A typica beam-coumn joint is iustrated in Figure 1b. The joint is ideaized as a threedimensiona eement, with dimensions d (width of coumn), b (width of beam) and h (height of beam). Average stresses in the joint are shown in Figure a,b. Shear stresses are introduced by direct member action and by bond that deveops between the main reinforcement and the joint core concrete. For simpicity, it is assumed that the shear stress, v, is uniformy distributed over the boundaries of the joint. Furthermore, it is assumed that at the moment of strengthening the joint is aready oaded, so that a set of initia norma strains, ε οt and ε ο in the transverse (beam) and ongitudina (coumn) direction, respectivey, and an initia shear strain, γ ο, have deveoped. N h v β t f t f t f ft σ t t N v v t (a) (b) f f f β f σ Figure. Stress equiibrium. (a) Horizonta forces; (b) vertica forces. The principa strains, ε 1 and ε, are reated to those in the ongitudina and transverse directions, ε, and ε t, through the foowing expression: tan ε1 εt θ = ε ε 1 ε ε = ε ε t (1) where θ = incination (from the t axis) of the maximum principa strain ε 1. Moreover, assuming that: (a) the maximum principa stress in the concrete, σ 1, cannot exceed the tensie capacity, which is taken zero; and (b) the directions of principa strains and stresses coincide (this is neary correct if the reinforcement has not yieded), one may show that: σ t = v ()

3 v σ = (3) where σ t and σ is the average compressive stress in the concrete in the transverse (t) and ongitudina () direction, respectivey. Furthermore, with σ 1 =, the stress invariant condition gives the minimum principa stress in the concrete: Finay, the shear strain γ in the joint equas: σ = σ t + σ () γ = ( ε ε ) ( ε ε ) 1 t = 1 tan θ t (5) Equiibrium Considerations A key assumption in this section is that the FRP materia consists of sheets or fabrics (with fibers in directions that do not coincide necessariy with the vertica and/or horizonta) that are stacked to form a aminate of thickness t f. In this case the reinforcement ratio in each direction is ρ ft = ρ f = ρ f = nt f /b, where n is the number of aminates (n = for two-sided jackets, when both sides of the joint are accessibe; n = 1 for one-sided jacket, when a transverse beam exists so that appication of the jacket on both sides is not possibe). Stresses and strains in the composite materia are couped according to the foowing constitutive aw: f f f ft f ft = ε t ε ot ε ε o γ γ o (6) where f ft = average norma stress in the FRP in the transverse direction (at mid-width of the joint); f f = average norma stress in the FRP in the ongitudina direction (at mid-height of the joint); f ft = shear stress in the composite materia; and ij (i, j = 1,, 3) are eements of the composite materia stiffness matrix that depend on the properties (four eastic constants and thickness) of the various aminae (ayers) that have been stacked to form the joint s externa reinforcement. Horizonta force equiibrium requires that σ t satisfy the foowing: Nh = ( ρs + βt ρb ) ft ρ f [ 11 ( εt ε ot ) + 1( ε ε o ) + ( γ γ o )] (7) bh σ t 13 where f t = average stress in the horizonta stirrups (at mid-width of the joint); ρ s = stirrup reinforcement ratio; ρ b = tota main beam reinforcement ratio; β t = factor with vaues between -1, reating the magnitude of stresses (or strains) in the main beam reinforcement to the average stirrup stresses (or strains) at the coumn centerine; and N h = compressive axia force of the beam (if any). The factor β t accounts for the bond conditions aong the main beam reinforcement: for perfect bond, β t =, for negigibe bond β t = 1 (Pantazopouou and Bonacci 199). Simiary, vertica force equiibrium gives the average ongitudina compressive stress in the concrete, σ, as foows: 3

4 ( ρ + β ρ ) f ρ ( ε ε ) Nv = c, in c f [ 1 t ot + ( ε εo ) + ( γ γ o )] (8) bd σ 3 where f = average stress of ongitudina reinforcement of the coumn inside the joint core at the mid-height of the joint; ρ c,in = coumn reinforcement ratio inside the joint core; ρ c = tota main coumn reinforcement ratio (at the boundaries of the joint core); β = factor that reates the magnitude of stresses (or strains) in the main coumn reinforcement to the average stresses (or strains) of the reinforcement inside the core at the beam centerine; and N v = compressive axia force of the coumn. As above, the factor β accounts for the bond conditions aong the main coumn reinforcement (at the boundaries of the core). To imit the number of variabes in the probem, we make the foowing simpifications: ρ t = ρ s +β t ρ b = effective horizonta reinforcement ratio, ρ = ρ c,in +β ρ c = effective vertica reinforcement ratio. Moreover, we assume that the effective yied stress of the horizonta reinforcement, f yt, is given as f yt = (ρ s f sy +β t ρ b f by )/ρ t, where f sy = yied stress of stirrups and f by = yied stress of beam reinforcement. The yied stress of the coumn reinforcement is denoted as f y. Next we anayze a the possibe states of joint behavior. Anaysis before yieding of stee We start with Equation 1 and the materia constitutive aws: tan ε ε σ σ θ = = ε εt (9) ε εt Ε c Ε c 1 where E c = is the secant eastic moduus of concrete in the strain under consideration. The stress σ is written in terms of v and using Equations -. The resuting expression is: 3 nscv vn sc tan θ εt Es tan θ + + ε tan θ E s () where n sc = E s /E c. Next we write Equation 7 with σ t repaced by v, γ repaced by the right term in Equation 5 and f t = E s ε t. The resut is obtained in terms of v as foows: 1 v = ρ Esεt + ρ f 13 + ε t + ρ f 1 tan θ 13 ε ρ f K 1 tan θ t Nh + bh (11) where K 1 11ε ot + 1ε o + 13γ ο = (1) Finay, we write Equation 6 with σ repaced by v/, γ repaced by the right term in Equation 5, f = E s ε and v as given in Equation 9. The resut in terms of ε is as foows: where ( ρ E + ρ A) ε ρ ( K K tan θ ) Nh Nv t s f t f 1 + tan θ ε = bh bd (13a) ρ B + ρ E tan θ f s

5 13 3 A = tan θ (1) 1 tan θ 1 tan θ 13 3 B = 1 + tan θ (15) 1 tan θ 1 tan θ Κ 1εot + εo + 3 = γ (16) Anaysis after yieding of effective horizonta reinforcement and before yieding of effective vertica reinforcement Τhe anaysis is carried out as in..1 above with f t = f yt. Hence v is given by Equation 11 with the product E s ε t repaced by f yt and the expression for ε becomes: ( K K tan θ ) Nh Nv ρ f Aεt ρ f 1 + tan θ + ρt f yt ε bd = bh (13b) ρ B + ρ E tan θ f s Anaysis after yieding of both horizonta and vertica reinforcement The anaysis is carried out as in..1 above with f t = f yt and f = f y. Hence v is given by Equation 11 with the product E s ε t repaced by f yt and the expression for ε becomes: ( K K tan θ) Nh Nv ρ f Aε t ρ f 1 + tan θ + ρt f yt ρ f y tan θ ε bh bd = (13c) ρ B f Anaysis after yieding of effective vertica reinforcement and before yieding of effective horizonta reinforcement Here too, the anaysis is carried out as in..1 above with f = f y. The shear stress v is given by Equation 11 and ε is: ( ρ E + ρ A) ε ρ ( K K tan θ ) Nh Nv t s f t f 1 + tan θ ρ f y tan θ ε bh bd = (13d) ρ B f Compressive crushing of concrete During any of the preceding states the concrete may crush; this wi define faiure of the joint. Crushing wi occur when the principa compressive stress, σ, reaches the strength of concrete, f c max. The stress-strain reationship assumed here aong the principa compressive direction is that described in Pantazopouou and Bonacci (199): ο max ε ε σ = f c (17) ε max ε max 5

6 max fc = λfc where ε max = λεο, 1+ ρsv f ys / fc λ =.8.3( ε / ε ) 1 ο f c and ε ο (= -.) are the compressive strength and faiure strain of concrete in uniaxia compression (they both carry negative signs) and ρ sv is the voume ratio of stirrups. Faiure of the FRP The FRP wi fai by tensie fracture when the tensie stress (f ft or f f ) reaches the tensie strength, f fu. Debonding is treated here according to the fracture mechanics-based mode of Hozenkämpfer (199), as modified by Neubauer and Rostásy (1997). This mode gives the maximum tensie stress in an FRP sheet of thickness t f when debonding occurs, f f,deb, in terms of the FRP eastic moduus parae to the oading direction, E f, the mean tensie strength of concrete, f ctm, and the bond ength, b. Soution procedure The anaytica formuation given above was impemented in a computer program that was specificay deveoped for the anaysis of RC joints strengthened with FRP sheets. The user inputs a series of materia and geometric characteristics and the program traces the state of stress and strain in the joint unti faiure. Input to the program consists of: (i) the geometric variabes ρ s, ρ b, ρ c,in, ρ c, ρ f ; (ii) the bond condition variabes β t and β ; (iii) the materia properties f c, f ctm, ε ο for concrete and E s, f y, f ys, f yb for stee; (iv) the geometric and eastic constants of the various aminae forming the FRP aminate (the program cacuates automaticay the eements ij of the FRP stiffness matrix) and the faiure criterion for fracture of the aminate; (v) the normaized axia forces N v /bd and N h /bh; and (vi) the initia strain ε ot in the joint (at the moment of strengthening). Note that for the most common case of aminates with fibers in the two orthogona directions ( and t), in (iv) above it is sufficient to define the utimate FRP stress, f fu,t in the direction t and f fu, in the direction. Upper imits to the FRP stress are aso introduced to account for debonding; these vaues are estimated as described in..6 using the approach described in the previous section with E f taken equa to 11 or, for the imiting vaues of f ft or f f, respectivey. As a first step, the program cacuates the initia strain ε o required to satisfy equiibrium of the joint (without the FRP). Next, the strain ε t is incremented and through an iteration scheme Equations, 11 and 13 are soved for, v and ε. The vaue of ε is aways obtained by soving the equation corresponding to the state that is active in each step. At the end of each step the program checks for FRP debonding or concrete crushing, which define the shear capacity v max (at east equa to that of the joint as if no FRP had been appied, v o,max ). In principe, E c is to be obtained through a secant moduus iteration scheme at each step. However, extensive anayses performed by the authors on FRP-strengthened joints as we as by Pantazopouou and Bonacci (199) on RC joints without FRP ed to the concusion that quite simiar resuts can be obtained without iteration by choosing E c to be the secant moduus at peak stress; this vaue may be assumed equa to f c /ε ο, that is E c = f c. 6

7 Numerica Study In the preceding sections agebraic expressions were derived for stresses and strains in RC joints strengthened with FRP materias at various states of the stee reinforcement (eastic, postyied). In this section the equations for joint shear strength are appied to a generic joint strengthened with fexibe sheets appied in severa ayers (aminae). The joint is assumed reinforced with a ot more reinforcement in the coumn than in the beam (ρ =.15, ρ t =.6). Each FRP ayer (amina) consists of unidirectiona carbon fibers in an epoxy matrix and has the foowing eastic constants: eastic moduus parae to the fibers E II = 18 GPa, eastic moduus perpendicuar to the fibers E = GPa, shear moduus G II, = 5 GPa and Poisson s ratio ν II, =.5. For the concrete we assume f c = 5 MPa and f ctm = 1.97 MPa. For the yied strength of the stee reinforcement we take f y = MPa and f yt = 3 MPa. Finay N h = and ε ot =.. v (MPa) v (MPa) N v /bd =.5 MPa 1. V = /1. V =.5/.5 3. V =.33/.67. V = 1/ 3 1 Contro specimen t-yied, -yied, crushing 3 t-yied, -yied, ε ft =.1 (a) t-yied, ε ft = N v /bd =.5 MPa 1. V = /1. V =.5/.5 3. V =.33/.67. V = 1/ ρ f Contro specimen t-yied, -yied, crushing t-yied, -yied, t-debond. 3 t-yied, -yied, ε ft =.1 t-yied, t-debond. t-yied, ε ft =.1 (b) ρ f 3 1 Figure 3. Shear strength of FRP-strengthened joint in terms of ρ f for various fiber distributions. t- yied = yieding of beam reinforcement, -yied = yieding of coumn reinforcement, t- debond. = debonding of beam reinforcement. If FRP debonding is not a concern, absoute dimensions of the joint need not be specified, as they serve ony to normaize stee/frp quantities and axia oads. But if debonding dominates, we 7

8 need to know t f, that is bρ f /n, and the FRP bond ength in each direction. Hence, for each ρ f we need to know the number of joint sides covered by the FRP (n = 1 or ) and the width of the beam, b. In this case study b = 5 mm and n =. The bond engths aong the t and directions are taken as bt = 5 mm and b = 5 mm. Next we define as V and V t the voume fraction (within the aminate) of ayers paced in the coumn and beam direction, respectivey (V + V t = 1). The foowing four configurations of the carbon sheets are assumed: (a) a ayers with the fibers in the direction of the beam, V /V t = /1; (b) the number of ayers with fibers in the beam direction is the same as that in the coumn direction, V /V t =.5/.5; (c) the ayers with fibers in the beam direction are two times more than those with fibers in the coumn direction, V /V t =.33/.67; and (d) a ayers with the fibers in the direction of the coumn, V /V t = 1/. From the initia strain ε ot before strengthening of the joints the other two eements of the initia strain matrix are cacuated as foows: ε o = -6.97x -5 and γ o = 1.95x -. Appication of the procedure described above gives the shear strength of the joint in terms of the amount of FRP as shown in Figures 3a, b for N v /bd =.5 MPa. Figure 3a appies if debonding is not of concern and Figure 3b appies with debonding taken into account. Each figure gives aso the state of reinforcement at faiure (stee may have yieded and the FRP may debond or fracture, at a hypothetica strain taken here equa to.1). The genera concusion is that if debonding is not an issue, the effectiveness (increase in shear capacity, v) of FRP is quite substantia and, for a given ρ f, it improves as more fibers are paced horizontay. This resut is not surprising, as for this particuar case study the joint stee reinforcement is much higher in the vertica direction than in the horizonta. If debonding is accounted for, the effectiveness of the FRP is reativey imited. Finay, FRP fracture is possibe at ow ρ f ony, and occurs in the horizonta (beam) direction. Experimenta verification Experimenta data on FRP-strengthened beam-coumn joints have been reativey imited. In order to vaidate the anaytica mode presented above and to obtain a more thorough understanding of the effect of various parameters on the behavior of RC joints, the authors conducted a comprehensive program that invoved simuated seismic testing of approximatey /3- scae T-joint modes. The joints were poory detaied (with no stirrups in the joint core) and the strengthening system was designed such that faiure woud occur due to shear. Earthquake oads were simuated by appying an aternating force (in a quasistatic cycic pattern) to the end of the beam through an ideaized pin and the axia force in the coumn was kept constant (Figure ). The dispacement-controed oading sequence for each specimen consisted of three cyces at a series of progressivey increasing (by 5 mm) dispacement ampitudes in each direction (push and pu), unti a dispacement of 5 mm was reached. Detais about these tests may be found in the recent artice of Antonopouos and Triantafiou (). From the oad versus dispacement curves (see Figure 5 for some typica ones) it was possibe to record the peak force, corresponding to joint faiure, and based on that to cacuate: (a) the tensie force T b in the main beam reinforcement (cacuated from cross section anaysis); and (b) the shear force V c at the coumn face. The quantity (T b V c )/bd gives the experimentay obtained vaue for the shear strength of the joint, v max, which may be compared with the prediction of the anaytica mode. Another set of simiar - in principe - test resuts avaiabe in the iterature is that of Gergey et a. (), who tested (differenty detaied, compared to the above specimens) T-joints strengthened with CFRP and cacuated the shear stress based on the experimentay measured oad appied at faiure of the joints. 8

9 P, δ hydrauic actuator beam hinge threaded rods (embedded in the concrete) oi jack coumn axia oad easto meric pad hinge Figure. Schematic view of test setup and geometry of specimens. Tabe 1. Design parameters of joints tested. Specimen f c (MPa) N v bd (MPa) (L/θ) t f (mm) ρ f (x -3 ) E II (GPa) AT 1 F11 F F1 F1 FW GL SF T-F33 T-FS 3 GPR / o, /9 o / o, /9 o / o, /9 o / o, /9 o / o, /9 o 5/ o, 5/9 o / o, /9 o 3/ o, 3/9 o / o, /9 o /5 o, /-5 o /5 o, /-5 o /5 o, /-5 o /5 o, /-5 o /5 o, /-5 o /5 o, /-5 o AT: Antonopouos and Triantafiou (); GPR: Gergey et a. (). Notation of specimens is as defined by those who conducted the tests. ρ s =.17, ρ sv =.3, f y = 65 MPa. 3 Strips paced on one side of the joint debonded we before the peak oad (strength) was reached and were ignored. L denotes the tota number of ayers on both sides of the joint at an ange θ from the horizonta. 9

10 Both sets of test data described above were used to evauate the proposed anaytica mode. A few test resuts were omitted from the comparison, because the associated strengthening designs were considered as either unsuccessfu or unreaistic: three joints in the study of Antonopouos and Triantafiou () were strengthened with stiff strips that debonded quite eary (before the peak oad was reached), whereas four joints in the study of Gergey et a. () were strengthened with unreaisticay ow quantities (resuting in extremey ow axia rigidity) of FRP. Detais about the design parameters of the joints compared are given in Tabe 1 and the comparison between anaytica and experimenta vaues for the joint shear strength is given in Tabe. Uness described differenty in Tabe 1, in a these tests ρ s and ρ c,in were equa to zero and the bond of rebars was assumed perfect, corresponding to β t = β =. The ast assumption was verified in the tests of Antonopouos and Triantafiou (), whereas no detais are provided by Gergey et a. () regarding rebar sip. P (kn) P (kn) P (kn) C1 (a) δ (mm) FW (c) δ (mm) T-C (e) δ (mm) P (kn) P (kn) P (kn) F11 (b) δ (mm) GL (d) δ (mm) T-F33 (f) δ (mm) Figure 5. Typica oad versus dispacement curves.

11 Tabe. Comparison of anaytica predictions with test resuts. Specimen AT F11 F F1 F1 FW GL SF T-F33 T-FS GPR v max Experimenta (MPa) v max Anaytica (MPa) v max -Ana. v max -Exp The authors found the agreement between anaysis and test resuts surprisingy good, and fee confident that the anaytica procedure deveoped in this study may be used as a vauabe too towards the design of FRP jackets for shear strengthening of beam-coumn joints. Concusions Anaytica modes are presented in this study for the anaysis of RC joints strengthened with composite materias in the form of externay bonded jackets comprising strips or fabrics with fibers in any direction. The modes provide equations for stresses and strains at various stages of the response unti the utimate capacity is reached, defined by concrete crushing or FRP faiure due to fracture or debonding. Soutions to these equations are obtained numericay. The modes provide usefu information on the shear capacity of FRP-strengthened joints in terms of the quantity and configuration of the externay bonded reinforcement and may be used to design FRP jackets for poory detaied beam-coumn joints. Parametric anayses indicate that even ow quantities of FRP materias may provide significant enhancement of the shear capacity. The effectiveness of externa reinforcement increases consideraby if debonding is suppressed (e.g. through proper anchorage) and depends heaviy on the distribution of ayers in the beam and the coumn. The atter depends on the reative quantities of stee reinforcement crossing the joint pane and the eve of axia oad in the coumn. Shear strength predictions provided by the anaytica modes were found in extremey good agreement with 15 experimenta resuts found in the iterature, thus adding confidence to the vaidity of the proposed equations. 11

12 Acknowedgements Partia support to this research has been provided by the Research Committee of the University of Patras ( Κ. Karatheodoris Program) and by the Genera Secretariat for Research and Technoogy (PENED 1999). References 1. Antonopouos, C.P. and Triantafiou, T.C. (), Experimenta Investigation of FRPstrengthened RC Beam-coumn Joints, Journa of Composites for Construction, ASCE, accepted.. Gergey, J., Panteides, C.P. and Reaveey, L.D. (), Shear strengthening of RC T- Joints using CFRP composites, Journa of Composites for Construction, ASCE, (), pp Hozenkämpfer, P., (199), Ingenieurmodee des verbundes gekebter bewehrung für betonbauteie, PhD dissertation, TU Braunschweig, Germany.. Neubauer, U. and Rostásy, F.S. (1997), Design Aspects of Concrete Structures Strengthened with Externay Bonded CFRP-Pates, In Concrete+Composites, Proceedings of the 7 th Internationa Conference on Structura Fauts and Repair,, pp Pantazopouou, S. and Bonacci, J. (199), Consideration of uestions about Beam-coumn Joints, ACI Structura Journa, 89(1), pp