CONSISTENT INSERTION OF BOND-SLIP INTO BEAM FIBER ELEMENTS FOR BIAXIAL BENDING

Transcription

1 CONSISEN INSERION OF BOND-S INO BEAM FIBER EEMENS FOR BIAXIA BENDING GIORIGO MONI AND ENRICO SPACONE 2 SMMARY In thi paper a new reinforced concrete beam finite element that explicitly account for the bondlip between the reinforcing bar and the urrounding concrete i preented. he element formulation combine the fiber ection model with the finite element model of a reinforcing bar with continuou lip. he ection model retain the plane ection aumption, but the teel fiber train are computed a the um of two contribution: the rebar deformation and the anchorage lip. he propoed finite element include material nonlinearitie in the concrete, teel and bondlip contitutive law. he formulation applie to both monotonic and cyclic load, and i therefore uitable for the eimic analyi or reinforced concrete tructure in which the rebar lip mut be accounted for. Since each longitudinal teel bar i monitored eparately, the model applie to any cro ectional hape and extend to both uniaxial and biaxial bending. Firt, the theoretical framework i preented, then a correlation tudy with an experimental tet on a cantilever column with circular cro ection i dicued. hi tudy how that the prediction with the new model i in good agreement with the experimental tet, while the original fiber model with perfect bond overetimate the hyteretic energy diipated during the loading cycle. INRODCION By modelling the repone of ingle embedded bar and r.c. member with bar anchored in the footing, it can be noticed that bond-lip affect the local behavior of the bar. A a conequence of thi local behavior, the bae moment-curvature diagram i equally affected, even though to a leer extent, but it can be oberved that the requet of ductility i lightly higher than in the cae with full bond. Minor difference are oberved in the forcediplacement diagram, becaue global quantitie are le enitive to local modification. Nonethele, alo in thi cae the initial tiffne i lower. hi may have meaningful effect on more complex tructural ytem, whoe repone to dynamic action i tiffne-driven. Notwithtanding the recognized importance of thee effect, in the analyi of reinforced concrete tructure, perfect bond i uually aumed between the rebar and the urrounding concrete. hi i true for low load level, wherea, a the load increae, cracking a well a breaking of bond unavoidably occur and bond-lip take place in the beam. Near the crack, high bond tree develop at the teel-concrete interface cauing relative diplacement between concrete and reinforcement. In particular, two effect are ignificant: a) an increae in tiffne in the region between two adjacent crack, and b) an increae of flexibility at the member end, due to the pullout of the rebar at the interface with either other framing element or the footing. Similar drop in tiffne may alo be caued by inufficient lap plice. hee effect become particularly important and complex under eimic loading condition, when bond deteriorate due to large train and damage caued by load reveral. he formulation of the beam element including the bond-lip effect derive from that originally propoed by Spacone et al. (996), with the inertion of the continuou bond element developed by Monti et al. (997). he framework of the fiber ection tate determination i retained, while a new approach i propoed for computing the rebar tre and tiffne that include the effect of lip. he repone of the bar i lumped at the fiber level. he teel fiber train i given by the um of the effect of the rebar deformation and the anchorage lip. he end 2 Aociate Profeor, niverità a Sapienza di Roma, Via Gramci, Rome, Italy - monti@uniroma.it Aitant Profeor, niverity of Colorado, Boulder, CO , SA - pacone@colorado.edu

2 lip i alo calculated, to give an etimate of the crack width. Alo the anchorage length outide the element, in either a tructural joint or a footing, i accounted for. hi make the new model particularly eay to implement into any exiting fiber beam finite element. hi approach allow to conitently account for bond-lip alo in biaxial bending condition. REINFORCED CONCREE FIBER BEAM EEMEN WIH BOND-S In fiber beam finite element, the tate determination i carried out at three level: element, ection and fiber. At all three level, the problem i the ame: determine the force (or tree) and tiffne correponding to precribed deformation. In particular, the ection tate determination compute the ection axial force and bending moment correponding to precribed ection deformation, namely the average axial train ε and the ection curvature κ. he dicuion i limited here to the uniaxial bending cae, but extenion to the biaxial cae i traightforward. From the plane ection aumption, the train at a fiber located at a ditance y from the reference axi i: ε = ε κy () When perfect bond i aumed between concrete and teel rebar, concrete and teel fiber located at the ame depth y have the ame train: hi aumption i removed in order to include the effect of bond-lip. ε = εc = ε = ε κy (2) Becaue of the numerical integration ued to olve the element integral, the beam element i modeled a a et of adjacent lice that are connected in erie. he element repone i the weighted um of the repone of each =. he lice lice. If one conider a beam element of length beam, the length of each lice i w beam length i a function of the integration cheme, the number of integration point and thu the weight w pertaining to the integration point. Each lice can be conidered a a parallel ytem made of two component, teel and concrete. From the aumption of perfect bond, the lice and it component have the ame elongation d and rotation ϕ, and ϕ ϕ d = dc = d ϕ = ϕ c = ϕ (3) in which dc d c, are the lice elongation and rotation in the concrete and teel component, repectively. he lice i baically a finite length of beam element with contant deformation. hee are the axial train ε and the curvature κ of the integration point, therefore: R S d = ε ϕ= κ With thi notation, the compatibility condition of Eq. (2) can be expreed in term of concrete and teel diplacement. he elongation of a fiber at ditance y from the reference axi can be written: (4) From Eq. () and (4) the train ditribution ε y bg= c = = ϕ (5) uy u u d y bg acro the monitored ection can be written: 2

3 εbg y = ϕ uy bg= cd yh Baed on thee definition, it i now poible to enhance the repreentation of the lice deformation by relaxing the aumption of perfect bond between teel and concrete. hi can be done by auming that the deformation of the teel component i due partly to the bar elongation and partly to the lip between the bar and the concrete. hi i expreed in the following form: (6) d = d = d = d d ϕ = ϕ = ϕ = ϕ ϕ c a a c a a (7) in which the lice axial deformation d and the lice rotation ϕ are expreed a um of two contribution, one due to the rebar deformation (ubcript ), the other to the anchorage lip (ubcript a). From Eq. (7), the concrete train i the ame a in Eq. (6) wherea the teel train become bg c h ε yc = d ϕ y = ε κ y c c c c (8) c h c a ayh (9) ε a= d ϕy d ϕ in which the firt term repreent the train of the teel bar in the lice and the econd i the contribution of the bond-lip. herefore ε a ε ε ε u = a = a () in which ε a hould be regarded a a train-equivalent contribution of the anchorage pullout, condened at the fiber level through the length of the integration point. In other word, the total teel fiber elongation i given by the um of the rebar deformation and anchorage pullout. Compatibility i maintained between the concrete train and the total teel fiber elongation ( εc = ε a), while concrete and teel train are different ( εc ε ). hi procedure i illutrated in Figure for the cae of a lice with axial deformation only and no curvature. he fiber ection tate determination i identical to that preented in Spacone et al. (996a). Given the ection bg= deformation ε and κ, find the correponding ection force and tiffne. he fiber train ε y ε κ y are computed firt. Baed on the new train field the fiber tree and tangent moduli are computed. hey are then integrated over the cro ection to yield the ection force and tiffne. he main difference introduced by the new approach concern the fiber repone. While the concrete train i directly computed from Eq. (8), the teel train determination i more involved becaue Eq. (9) yield the total teel repone ε a. A pecific procedure ha been developed to compute the rebar deformation ε and average anchorage lip ε a correponding to ε a. 3

4 Slice length Cro Section axial train = ε curvature = ε = lice elongation Element with perfect bond ε = ε c = ε Element with partial bond ε = ε c = ε ε a ε a (ε ) Figure. Slice repone to axial deformation only: perfect bond v. partial bond Steel Bar State Determination he teel fiber tate determination compute the rebar tre and tiffne correponding to the total train ε = ε u. he ytem repreent a teel rebar of length plu the bar anchorage of length a. In a a thi cae the anchorage length a extend outide the element in the beam-column joint. he rebar i modeled with a imple bar element, whoe contitutive law i that of a reinforcing bar. he embedded bar i modeled by a erie of rebar with continuou bond embedded in concrete, following the approach propoed by Monti et al. (997a and b). he following linearized contitutive law are ued: σ =E ε R S σ k u a aa an a W = N M O kna knn Q P R S un VW () V in which the anchorage dof have been plit into the anchorage diplacement u a and the diplacement u n of the other n node along the anchorage length. he vector in the tre vector indicate that the bar i ubjected only to an end tre σ a, while all other dof have no applied tre. he lat node n at the anchorage tip i therefore unretrained. he value of n depend on the number of element ued to dicretize the length a of the anchorage zone. A tudy on the optimal number of element can be found in Monti et al. (997a). he following deformation vector e i aociated with each fiber k e = lε ua unq (2) ing thi notation the train in the teel fiber anchorage ytem (a) given by Eq. () i written in which m = l ε a q. A teel fiber tre vector i alo defined = m e (3) = mσ σa nr (4) 4

5 in which n i the tre unbalance vector at the node along the anchorage length. hi unbalance i included becaue of the nonlinearity of the problem. During the olution proce, it i likely to have n. When the anchorage olution i reached n. Due to the erie arrangement of teel and anchorage expreed by Eq. () the teel fiber and the anchorage have the ame applied tre which allow rewriting the tre vector of Eq. (4) a σ a= σ= σa (5) σ a n = m he two local contitutive relationhip () are grouped in a ingle expreion = K e = NM m r (6) E kaa k an k k na nn O QP (7) he tiffne matrix in Eq. (7) can be built uing the virtual force principle. he virtual force principle can then be written in the following form: e δσ ε = δ e (8) he virtual variation of the tre vector hould be in equilibrium ( δ n = ), and i given by a a δ = m δσ a (9) pon ubtitution of (9) into (8) and after elimination of the virtual variation δσ a from the tandard argument of arbitrarine, Eq. (8) i written ε a = = = m e m K m F in which Eq. (7) ha been ubtituted. he flexibility matrix F i given by F = K = pon ubtitution of (6) into (2), one obtain and Eq. (22) i inverted to yield ε NM E faa fan f f na = m F m σ m F a a nn R S O QP n V W (2) (2) (22) 5

6 σ = E ε ε a a a n in which E a i the tiffne of the teel plu bond fiber and the train E a = m F m = E ε n = R S m F n c h (23) F HG V W = repreent the reidual train due to the unbalanced force along the anchorage length. A the olution approache convergence, ε n. he reult of thi procedure i chematically hown in Figure 2. f an f n aa I KJ (24) (25) Beam Cro Section Beam Element State Determination of Integration Point a ndeformed Integration Point Concrete (Steel and Anchorage) (ε ) u n u u a u n Deformed Integration Point: Concrete op (S A) only ε = ε u a a Figure 2 Steel bar tate determination including anchorage lip COMPARISON WIH EXPERIMENA RESS In thi ection a comparion with an experimental tet performed by Saadatmaneh et al. (996) i performed. he tet wa conducted on a caled-down circular reinforced concrete bridge column, deigned according to obolete eimic code, with inufficient development length for the rebar anchored in the footing. Readymixed concrete with f c = 36.5 MPa and teel with meaured yield trength of 358 MPa were ued. he diameter of the column wa 35 mm and the longitudinal reinforcement wa 4 bar of 3-mm diameter, reulting in a reinforcement ratio of 2.48%. ranvere confinement wa provided by teel wire hoop of diameter 3.5 mm and paced at 89 mm throughout the entire height of the column. he average yield tre for thee wire wa 3 MPa. In the pecimen (denoted a C-) the longitudinal reinforcement of the column wa extended into the footing uing tarter bar that were lapped with the main longitudinal reinforcement of the column over a length of 2 bar diameter (254 mm). In the tet an axial load of 445 kn wa firt applied to the column, followed by cyclic lateral diplacement applied at the top of the column in both poitive and negative direction according to the cheme: u y,.5u y, 2u y, 3u y, where u y i the yield diplacement. At each diplacement 6

7 level, two cycle were performed. ure 3a how the hyterei loop for the experimental repone of the circular column C- with lap plice. (Saadatmaneh et al. (996)). he weak bond repone already oberved in the previou ection characterize the tet. After the firt cycle to u=.5u y, there i a rapid degradation of the repone due to progreive failure of the lapped reinforcement due to lip. In addition, it i noted that the cycle how ignificant pinching, with extremely reduced diipation capacity. ure 3b preent two numerical imulation of the experimental tet. he firt wa performed with the fiber element model with perfect bond preented in Spacone et al. (996a), the econd ue the model propoed in thi paper that include bond-lip of the rebar. he difference in the prediction are remarkable, the mot evident regarding the diipation capacity, which i of paramount importance in eimic analye. he model with no bond-lip predict fuller cycle, which are far from being imilar to thoe obtained in the tet. he prediction obtained with the new model i imilar to the experimental reult. Another remarkable difference regard the prediction of the peak load cyclic degradation, he repone obtained with the new model how a decay that i very imilar to the experimental cae, thank to it capacity of tracing the bond-damage penetration into the anchorage. A more ophiticated cyclic degradation rule in the bond-lip contitutive law would lead to further enhancement in the prediction capabilitie of the propoed model. Finally, it i important to point out the complexity of the imulation in which the reinforcing bar all have different repone, ince the teel i ditributed along the perimeter of the circular reinforced concrete ection and all the rebar have different train. OAD (kip) SPECIMEN C- FE with bond-lip FE with no bond-lip Serie3 Serie AERA DISPACEMEN (in) Figure 3 Comparion between experimental tet (left) from Saadatmaneh et al. (996) and numerical imulation (right) with and without bond-lip CONCSIONS A fiber beam element that include bond-lip of the reinforcing bar ha been preented. he formulation i derived from the force-baed fiber beam element propoed by Spacone et al. (996a), with the inertion of bar element with continuou bond developed by Monti et al. (997a and b). he repone of the beam element i obtained through the weighted integration of the repone of the monitored ection along the element. he fiber ection repone i computed by adding the contribution of all the concrete and teel fiber in which the ection i ubdivided. In the propoed beam element, the teel fiber tate determination i modified to account for the bond-lip of the rebar. In the original model, the teel tre and tiffne are calculated directly from the teel contitutive law, baed on the train pertaining to the fiber. In the new model, the tre and tiffne of each teel fiber are obtained a the repone of a bar that croe the ection under conideration and i anchored in the concrete. he end lip can alo be computed, in order to give an etimate of the crack width. he implementation i imple and require minor modification to the tructure of the original fiber beam element, provided a modular organization of the element routine i followed. Since the new formulation affect the fiber ection model only and doe not affect the overall element tate determination, the propoed fiber ection model with bond-lip in the teel rebar can be implemented in any diplacement-baed or force-baed that ue a fiber ection dicretization. In the preent form the model i implemented in a force-baed element that i linked to the general purpoed Finite Element program FEAP (aylor, 998). 7

8 Becaue each teel rebar in the cro ection i monitored eparately, the propoed model applie to cro ection of any hape. For the ame reaon, the model can be applied to both uniaxial and biaxial bending problem. A comparion between experimental data and numerical imulation ha been preented. hi tudy ha hown that the reult obtained with the propoed model correlate well with the experimental data, while the original fiber model with perfect bond largely overetimate the hyteretic energy diipated during the loading cycle. he propoed model can be applied to the tudy of reinforced concrete frame. For new tructure it will provide a more realitic prediction of the initial tiffne than that obtained from the fiber model with perfect bond. For older tructure with inufficient anchorage length or lap plice it can realitically account for bond deterioration and failure under both tatic and dynamic loading condition. Cae will be examined, where the anchorage length i inufficient, uch a beam-column joint. hi will be the bai for further tudie on complex tructural ytem where the lippage between two framing element will be accounted for. REFERENCES Monti, G., Filippou, F.C., and Spacone, E. (997a). Finite element for anchored bar under cyclic load reveral. Journal of Structural Engineering, ASCE, 23(5), Monti, G., Filippou, F.C., and Spacone, E. (997b). Analyi of hyteretic behavior of anchored reinforcing bar. ACI Structural Journal, May-June, 94(3), Saadatmaneh, H., Ehani, M.R., and Jin,. (996). Seimic trengthening of circular bridge pier model with fibre compoite. ACI Structural Journal, Nov-Dec, 93(6), Spacone, E., Filippou, F.C., and aucer, F.F. (996a). Fibre beam-column model for nonlinear analyi of R/C frame. Part I: formulation. Earthquake Engineering and Structural Dynamic, 25, Spacone, E., Filippou, F.C., and aucer, F.F. (996b). Fibre beam-column model for nonlinear analyi of R/C frame. Part I: application. Earthquake Engineering and Structural Dynamic, 25, aylor, R.. (998). FEAP: A Finite Element Analyi Program. er Manual: Verion 7. Department of Civil and Environmental Engineering, niverity of California, Berkeley. 8