MAXIMUM BENDING MOMENT AND DUCTILITY OF R/HPFRCC BEAMS

Transcription

1 MAXIMUM BENDING MOMENT AND DUCTILITY OF R/HPFRCC BEAMS Aleandro P. Fantilli 1, Hirozo Mihahi 2 and Paolo Vallini 1 (1) Politecnico di Torino, Torino, Italy (2) Tohoku Univerity, Sendai, Japan Abtract The train of high performance fiber reinforced cementitiou compoite (HPFRCC) at maximum tenile tre i higher than the teel train at yielding. For thi reaon, in reinforced HPFRCC (R/HPFRCC) tructure train incompatibility between rebar and cement-baed compoite doe not appear. A a conequence, the maximum bending moment of R/HPFRCC beam can be eaily evaluated by mean of the uncracked formula (or Stage I formula). But, unlike the bearing capacity, in the evaluation of deformability, train localization ha to be taken into account during failure. In order to highlight the importance of localization, numerical reult and experimental meaurement are compared in the preent paper. Thi can be done by introducing a mechanical model for the evaluation of the moment-curvature relationhip, and by referring to a tet on a R/HPFRCC beam in three point bending. If train localization in the compreed HPFRCC i correctly modelled, a good agreement between the reult of the propoed model and the experimental data i found. 1. INTRODUCTION High performance Fiber Reinforced Cementitiou Compoite (HPFRCC), introduced in the market of cement-baed compoite in the lat decade, can be conidered a a pecial family of Fiber Reinforced Concrete (FRC). According to the definition of Naaman and Reinhardt [1], the high performance i related to the pot-cracking tage of tenile element, during which train hardening, or peudotrain hardening, appear in the tretrain (σ-ε) relationhip. By increaing tenile load, HPFRCC uually how a high number of crack of reduced width, without any train localization. Unlike ordinary concrete or FRC, a HPFRCC can utain tenile tree in correpondence of high train, although the fiber volume content doe not exceed 2%. Sometime, tree can be detected for train higher than thoe of teel at yielding. In other word, in the tenile zone

2 of a teel reinforced HPFRCC tructure (R/HPFRCC), the train of the cement-baed compoite are totally compatible with thoe of rebar. Confirmation of thi can be found in [2], where numerical and experimental analye of R/HPFFRC member in tenion have been conidered. If train compatibility exit, the tructural repone of R/HPFFRC tie, and the behaviour of reinforced beam with a HPFFRC layer in the tenile zone [3], can be conequently defined by the Stage I model (i.e. without conidering the bond-lip mechanim between rebar and cement-baed compoite). Recently, the behaviour of teel reinforced tructure totally made by HPFRCC ha been invetigated in everal experimental analye. In uch tet, R/HPFRCC element how higher performance than thoe of claical reinforced concrete (RC) tructure. In Naaman and Chandrangu [4], for intance, a R/HPFRCC lab i conidered in an innovative bridge deck ytem. Depite the reduction of teel reinforcement (about 70% lower than that of ordinary RC deck), the increae of bearing capacity, the increae of ductility (and of the platic rotation), and the decreae of the maximum crack width are clearly evident. Moreover, due to the ductility of HPFRCC, the ize effect hown by R/HPFRCC in three point bending i remarkably reduced [5]. To better undertand all thee phenomena, in the preent paper, the trength and the deformability of R/HPFRCC beam are invetigated. Preciely, by conidering the reult of a preliminary tet performed at the Tohoku Univerity of Sendai, a mechanical model i propoed, in order to reproduce theoretically the mechanical behaviour of a beam in three point bending. 2. TEST ON A R/HPFRCC BEAM IN THREE POINT BENDING The geometrical propertie of the R/HPFRCC beam teted in three point bending are reported in Fig. 1a. In the ame picture, the tet equipment and the poition of the intrument are alo illutrated. The load P ha been applied by mean of a hydraulic jack connected to a 30 kn load capacity Intron machine. The tet ha been controlled by the midpan vertical diplacement η of the beam, which ha been et up at maximum velocity of 0.2 mm/min. Thi deflection ha been evaluated by mean of the tranducer T3 in Fig. 1a. Two tranducer (T1 and T2 in Fig. 1a) have been alo aligned horizontally in a zone of length L t = 200 mm around the axi of the beam. In thi way, it wa poible to meaure the rotation ϕ and the average curvature µ av near the point of load application. 2.1 Material propertie The HPFRCC of the beam i compoed of cement, ilica fume, and, uperplaticizer, teel cord (1% in volume), and Polyethylene fiber (1% in volume). One teel rebar, named a SD295A D10 (Japan Standard Aociation), ha been adopted a reinforcement of the teted beam (Fig. 1a). The tre-train diagram of the material, obtained from a tet on HPFRCC cylinder in compreion (Fig. 1b) and in tenion (Fig. 1c), and from a tet on teel rebar in tenion (Fig. 1d), are alo reported in Fig. 1. In thee diagram, the dahed line repreent the contitutive relationhip adopted in the theoretical analye. In particular, for the HPFRCC in compreion, the σ c -ε c curve can be divided into two part. The acending branch can be reproduced by

3 the Sargin relationhip of CEB-FIP MC90 [6], while an exponential law i here adopted for the pot-peak branch (Fig. 1b): 2 f c [ Ec Ec1 ε c ε c1 ( ε c ε c1 ) ] [ 1 + ( Ec Ec1 2) ε c ε c1] for ε c ε c1 σ c = (1) f c - ( σ cu f c ) { 1- exp[ - 0.8( ε c ε c1 ) ε c1] } for ε c1 < ε c ε cu where E c and E c1 are evaluated with CEB-FIP MC90 formulae [6]. According to Kanda et al. [7], the pre-peak behaviour of HPFRCC in tenion can be decribed by the following bi-linear σ ct -ε ct relationhip (Fig. 1c): σ ct1 ε ct1 ε ct for 0 ε ct < ε ct1 σ ct = (2) σ ct1 + ( σ ct2 σ ct1) ( ε ct2 ε ct1) ( ε ct ε ct1) for ε ct1 ε ct < ε ct2 Finally, regarding the rebar, the acending branch of teel in tenion i well approximated by a tri-linear σ -ε relationhip (Fig. 1d): f y ε 1 ε σ = f y f y + u ( f f ) ( ε ε ) ( ε ε ) y [ ] 2 u for for for 0 ε < ε ε ε < ε 2 ε ε < ε u (3) Figure 1: a) Tet on a three-point bending beam; b-d) tre-train relationhip of material

4 3. MODELLING RC AND R/HPFRCC THREE-POINT BENDING BEAMS To carry out the tructural analyi of a beam in three-point bending, the croectional moment-curvature relationhip M-µ mut be defined. With reference to the generic cro-ection depicted in Fig. 2a, if the tre-train relationhip of the material and the value of the normal force N are known, under the hypothei of linear train profile (Fig. 2b), the M-µ diagram can be evaluated numerically. Figure 2: a-b) Strain profile in RC and R/HPFRCC beam; c) cruhing of concrete In particular, for a given value of the cro-ectional curvature µ, the train parameter λ (Fig. 2b), and the correponding value of the bending moment M, can be obtained by olving the following ytem of equilibrium equation (N=0 for the beam of Fig. 1a): B B H / 2 H / 2 H / 2 σ σ H / 2 ( y, ε ) dy = 0 ( y, ε ) y dy = M where the tate of tre σ(y) can be calculated through the σ-ε relationhip of Fig. 1b-d (Eq 1-3) by impoing the condition of linear train profile ε(y) = λ+µ y. It mut be remarked that the olution of the nonlinear ytem (4) (i.e. the cro-ectional M-µ diagram depicted in Fig. 3a) i not ufficient by itelf to define the tructural repone of a three point bending beam. Thi i due to everal phyical phenomena which affect a wide portion of RC and R/HPFRCC beam in bending. Only if the M-µ relation repreent an etimate of the average deformability of an entire block, can the momentcurvature relationhip be taken a the appropriate tool for the tructural analyi. Suffice to conider to the bond-lip mechanim between teel and cracked concrete in tenion, which i the well-known tenion-tiffening phenomenon [8]. It regard wide zone between crack, where, due to the concrete in tenion, the M-µ diagram appear tiffer (4)

5 than that conventionally computed by mean of the Stage II cro-ectional model (i.e. no tre in tenile concrete). However, a mentioned before, no train localization appear in HPFRCC in tenion, thu the bond-lip mechanim doe not affect the mechanical repone of R/HPFRCC tructure [2]. On the contrary, during failure, wide block of RC and R/HPFRCC beam can be intereted by other phenomena. Figure 3: a) M-µ diagram; b-c) M(z) and µ(z) ditribution in a beam in three-point bending In everal tructure, able to how ignificant ductile behaviour, platic hinge can be put in the cro-ection where the maximum bending moment M max ha been reached (Fig. 3a). In thi way, although heavy rotation, between the part connected by the hinge, are allowed, M max i kept contant in the cro-ection. Only when the moment-curvature relationhip can be decribed by monotonic function (or, a reproduced by the dahed line in Fig. 3a, M= M max for µ µ max ), can the platic hinge be conidered a a good approximation of the tructural behaviour. Thi i not true when material how train oftening propertie. In uch a ituation, both the M-µ diagram and the platic hinge exhibit oftening (that i, decreae of bending moment M with increaing curvature or rotation, repectively). Thu, intead of a ingle cro-ection, the moment-curvature diagram hould be referred to a wide block of the beam. The length L c of thi block, the o-called oftening region [9], cannot be arbitrary, and in particular it mut be larger than zero. In a firt intance, the condition L c > 0 hould be introduced in order to avoid the o-called Wood-Robert paradox [10]. It appear when a cro-ectional M-µ relationhip i conidered in the tructural analyi, and the beam i divided into block of infiniteimal length. A Fig. 3b how, after reaching the maximum bending moment M max in the midection, an increment of curvature in thi ection caue a reduction of the bending moment. On the contrary, in the cloet ection both bending moment and curvature decreae (Fig. 3a-b). In other word, in the zone urrounding the

6 cro-ection at z = L/2, when L c 0 a dicontinuity appear in the curvature ditribution µ(z). Moreover, if a uitable length for the oftening region i not aumed (i.e. L c = 0), the energy diipation during oftening vanihe, which i not admiible (ee [9] for a review). When oftening in the M-µ relation i related to an appropriate value of L c, all the problem previouly mentioned can be avoided, and the M(z) and µ(z) ditribution can be correctly evaluated (Fig. 3c). In particular, during the oftening, bending moment and curvature are aumed to be contant within L c. However, the definition of L c appear intricate. In the cae of the R/HPFRCC three point bending beam depicted in Fig. 1a, ince hear failure ha not been oberved in the experimental analyi, oftening in the M-µ diagram can be acribed to the train oftening of the HPFRCC in compreion. A i well known, a oon a the train ε c1 i reached in the σ c -ε c relationhip (Fig. 1b), cruhing of compreed HPFRCC tart [11]. In thi ituation, the hypothei of localied damage around liding plane, inclined of α repect to the horizontal line, hould be aumed (Fig. 2c). The value of α can be numerically computed according to the Mohr- Coulomb failure criterion. Preciely, α = 17.5 i obtained by auming the ratio σ ct2 / f c equal to one tenth. Thu, by conidering x p a the depth of the compreed zone in which ε c > ε c1, the value of L c i computed by mean of the following equation: Lc = 2 x p / tanα (5) Since the length of the oftening region increae with the increae of beam deflection, M(z) and µ(z) ditribution can be evaluated with a numerical procedure. In particular, the value of x p i continuouly updated in the block where µ > µ max (Fig. 3ac). For a given value of P, the numerical procedure furnihe a more reliable ditribution of curvature µ(z), from which the average curvature µ av (referred to the block of length L t in Fig. 1a) and the midpan deflection η (Fig. 3c) are eaily computed. 4. COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL DATA Referring to the R/HPFRCC beam of Fig. 1a, the effectivene of the propoed approach can be checked by comparing the P-η and the M-µ av curve experimentally meaured with thoe theoretically computed. For a beam in three point bending, in order to highlight the apect of practical interet, the comparion regard eparately the bearing capacity and the evaluation of deformability (i.e. the ductility of the beam). 4.1 The maximum bending moment of R/HPFRCC beam The maximum bending moment M max of the beam depicted in Fig. 1a can be obtained from the cro-ectional moment-curvature relationhip, without conidering any train localization. In particular, the column defined by the acronym Num_1 in Fig. 4a, repreent the value of M max computed by adopting the contitutive relationhip of Fig. 1b-d. In the ame picture, the column of M max experimentally meaured (Tet in Fig. 4a) and numerically computed in two different analye (Num_2 and Num_3 in Fig. 4a) are alo reported. In the analyi Num_2, M max ha been obtained with a cro-

7 ectional model by adopting the tre-train diagram of an ordinary concrete in compreion (evaluated according to CEB-FIP MC90 [6] and the parameter reported in Fig. 1b), and the tre-train diagram of the HPFRCC in tenion (Fig. 1c). On the contrary, tre-train diagram of an ordinary concrete both in compreion (evaluated according to [6]) and in tenion are conidered in the analyi named a Num_3. In the bar chart of Fig. 4a, the dahed line identify the moment M u, and M y,, which are evaluated according to the platic deign method. In particular, M u, ha been obtained by conidering σ c = f c, σ ct = σ ct2 and σ = f u, wherea for M u,y the tate of tre defined by σ c = f c, σ ct = σ ct2 and σ = f y (Fig. 1b-d) i aumed. The bearing capacity of a R/HPFRCC cro-ection in bending can be accurately defined by the platic deign approach (in particular M u,y ) and by the numerical procedure correponding to Num_1 and Num_2, where the contribution of HPFRCC in tenion i taken into account. If tenile tree in the cement-baed compoite are neglected, a in the cae Num_3, M max i underetimate of about 30% with repect to the value meaured in the tet. Figure 4: a-d) Comparion between numerical reult and experimental data

8 4.2 Deformability of R/HPFRCC beam A previouly remarked, a uitable length L c for the oftening region hould be introduced in order to compute entirely the deformability of the beam depicted in Fig. 1a (that i the P-η and the M 0 -µ av diagram -where M 0 i aumed to be the maximum bending moment of the beam). At the moment, ince few data for the cruhing of compreed HPFRCC are available, the ame aumption made for ordinary concrete are here conidered [11]. In particular, the hypothei of localied damage around liding plane inclined of α = 17.5 i adopted. In Fig. 4b-c, the reult obtained with the propoed model are compared with thoe meaured in the tet. Only by taking into account L c > 0 in the propoed approach, do the theoretical reult follow the experimental pot-peak tage of P-η (Fig. 4b) and M 0 -µ av (Fig. 4c). Moreover, to better reproduce the oftening branche, a particular tre-train relationhip for the HPFRCC in compreion hould be adopted. A a matter of fact, the pot-peak σ c -ε c relationhip propoed by CEB-FIP MC90 [6] appear inadequate. Even if in the HPFRCC the length of oftening region appear horter than in ordinary concrete, at ame curvature µ av (Fig. 4d), ome other tet are needed in order to evaluate cruhing of HPFRCC in compreion. However, both in the erviceability tage (ruled by the ductility of the cementitiou compoite in tenion) and in the ultimate tage (ruled by the ductility of the cementitiou compoite in compreion) R/HPFRCC element in bending how higher trength and ductility than thoe meaured in ordinary RC beam. 5. CONCLUSIONS Due to train compatibility between teel rebar and HPFRCC in tenion, in the bearing capacity of reinforced beam, made totally or partially by HPFRCC, the contribution of the tenile tree in the compoite hould be taken into account. Neverthele, train compatibility i not ufficient by itelf to reproduce the deformability of R/HPFRCC beam in three point bending. A hown by the comparion between numerical reult and experimental meaurement, an appropriate length for the oftening region hould be introduced. In thi way, it i poible to imulate correctly the pot-peak branch of the load deflection curve. Although atifactory reult have been obtained, it i deirable that further invetigation are performed in order to better analye cruhing of HPFRCC in compreion. REFERENCES [1] Naaman, A.E. and Reinhardt, H.W., 'Characterization of high performance fiber reinforced cement compoite', in 'High Performance Fiber Reinforced Cement Compoite 2' (RILEM Proceeding 31, Ed Naaman and Reinhardt, E & FN Spon, London, 1996) [2] Fantilli, A.P., Mihahi, H. and Vallini, P., 'Strain compatibility between HPFRCC and teel reinforcement', Material and Structure 38 (2005) in pre. [3] Maalej, M. and Li, V.C., 'Introduction of Strain-Hardening Engineered Cementitiou Compoite in Deign of Reinforced Concrete Flexural Member for Improved Durability', ACI Structural Journal 92(2) (1995)

9 [4] Naaman, A.E. and Chandrangu, K., 'Innovative Bridge Deck Sytem Uing High- Performance Fiber-Reinforced Cement Compoite', ACI Structural Journal 101(1) (2004) [5] Lepech, M. and Li, V.C., 'Size effect in ECC tructural member in flexure', in 'Fracture Mechanic of Concrete Structure', Proceeding of the V International Conference, Vail, Colorado, USA, April, 2004 (Ed Li at al., Ia-FraMCo, 2004) [6] CEB, 'CEB-FIP Model Code 1990', (Thoma Telford, London, 1993). [7] Kanda, T., Lin, Z. and Li, V.C., 'Tenile Stre-Strain Modelling of Peudotrain Hardening Cementitiou Compoite', ASCE Journal of Material in Civil Engineering 12(2) (2000), [8] Fantilli, A.P., Ferretti, D., Iori, I. and Vallini, P., 'Flexural Deformability of Reinforced Concrete Beam', ASCE Journal of Structural Engineering 124(9) (1998) [9] Bažant, Z.P. and Cedolin, L., 'Stability of Structure: elatic, inelatic, fracture, and damage theorie', (Oxford Univerity Pre, Inc., 1991). [10] Maier, G., 'Untable Flexural Behaviour in Elatoplatic Beam' (in Italian), Rendiconti Itituto Lombardo (A) 102(412) (1968) [11] Fantilli, A.P., Ferretti, D., Iori, I. and Vallini, P., 'Mechanical Model for the Failure of Compreed Concrete in Reinforced Concrete Beam', ASCE Journal of Structural Engineering 128(5) (2002)