Method for Determination of Tensile Properties of Engineered Cementitious Composites (ECC)

Transcription

1 Method for Determination of Tenile Propertie of Engineered Cementitiou Compoite (ECC) L. Øtergaard, R. Walter and J. F. Oleen Technical Univerity of Denmark, Denmark Abtract Today, ECC ha been developed to a tage where the material i being ued in eld teting and limited full cale contructed facilitie. However, the lack of reliable, practical, tandardized tet method for determination of the train hardening propertie of the material contitute a ignicant obtacle for day-to-day tructural engineering deign and material quality control during contruction execution. In practice a imple and robut tet method i required on a contruction ite. The uniaxial tenion tet (UTT) i regarded too complicated to erve a uch. Intead, the widely adopted four point bending tet (FPBT) may be ued for thi purpoe, provided that an appropriate interpretation of the tet data i available. The preent paper preent a contribution in thi direction. Sectional analyi of ECC beam in the train-hardening tage i ued in a parametric analyi aimed at revealing the ne detail of correlation uniquene between the tenile tre-train (σ-ε) curve parameter of an ECC and it FPBT load-deection diagram. Further, an invere analyi deignated for the determination of the train hardening propertie i decribed. Finally, experimental reult baed on FPBT tet reult are preented. Keyword: ECC, FPBT, invere analyi, hinge model, train hardening, tenile propertie Lennart Øtergaard Department of Civil Engineering Technical Univerity of Denmark DK-2800, Kg. Lyngby Denmark lo@byg.dtu.dk Tel.: (+45)

2 1. Introduction Over the pat decade, a new cla of civil engineering material ha been developed, often referred to a High Performance Fiber Reinforced Cementitiou Compoite (HPFRCC). A ubcla i the Engineered Cementitiou Compoite (ECC), which are engineered by detailed deign of the microtructure in order to obtain extraordinary toughne and ductility. The fundamental dierence between ECC and a conventional Fiber Reinforced Concrete (FRC) i it behavior in tenion. For FRC, fracture localization occur immediately after the rt crack i formed. In the cae of ECC, propagation of the initial crack will be prevented by the ber bridging, cauing a econd matrix crack to form elewhere. Thi reult in non-localized multiple cracking, alo known a macrocopic peudo train hardening behavior. ECC have preented new alternative in the deign of civil tructure. Due to their high ductility they are well uited in the deign of tructure expoed to cyclic loading, e.g. bridge lab and retrot of wall ubject to eimic loading. A comprehenive ummary of the tate of the art and preent application are given in [1]. The aforementioned paper, [1], point out that depite an increaingly complete material databae i being etablihed for the propertie of ECC under variou mechanical loading, there i till a need to develop tandardized tet method, particularly for the fundamental uniaxial tenile σ-ε curve. Such a method could greatly eae comparion between dierent ECC material which, today, relie on the complicated UTT. Alo in practice a imple and robut tet method i required for on-ite quality control and for material upplier. Formulation of a imple tandardized tet i one of the objective of a technical committee recently etablihed under the aupice of RILEM (RILEM TC HFC). Today, mot characterization of the σ-ε relationhip of ECC i carried out uing the UTT a hown in [2]. However, the UTT i generally regarded a too demanding with repect to laboratory equipment and ta expertie. Firt of all, the teting machine need to be cloed-loop controlled and uciently ti. Furthermore, no notch can be introduced into the pecimen, and thu, the experiment i particularly enitive to tre concentration occurring at the xation point of the pecimen. Alo the pecimen dimenion mut be oberved carefully, ince hrinkage or unprecie form lling may lead to tre variation throughout the pecimen, and thu, premature crack localization. A impler alternative to the UTT i the FPBT. Thi etup utilize that the moment i contant between the loading point, which allow for train hardening in the material and ubequent, determination of the σ-ε relationhip by invere analyi of the load-deection repone. Furthermore, the FPBT i already widely adopted in the uer community of cementitiou material [3], while invere analyi ha recently been attempted [4]. Thi paper explore the poibility of employing the FPBT for determination of the σ-ε relationhip of ECC. Sectional analyi of ECC beam in the train-hardening tage i ued in a parametric analyi aimed at revealing the correlation uniquene between the tenile σ-ε curve of an ECC and it FPBT load-deection diagram. Further, an invere analyi deignated for the determination of the σ-ε relationhip i decribed. The complete method i demontrated through the analyi of experimental FPBT reult.

3 ε ε σ ε ε ε σ 2. Modelling of ECC The peudo-train hardening propertie of ECC in tenion are often decribed uing a bilinear σ-ε relationhip ince thi aumption capture the main characteritic of the material behavior. Thi alo hold true for the compreive propertie, ee e.g. [5, 6]. Alo the poibility of cruhing of the ECC in the compreive region wa modelled in [5]. Thee contitutive condition are all included in the preent model and illutrated in Figure 1. f t2 f t1 1 a 1 f t2 c2 c1 t1 unloading branch t2 b 2 f t2 w 1 a 1 2 f t2 w 2 w f c1 f c2 (a) (b) Figure 1: Aumed contitutive relationhip for ECC: bilinear in tenion and compreion prior to crack localization (a), while a bilinear σ-w relationhip after crack localization i aumed (b). However, alo the localization and oftening of ECC i important when a FPBT i conidered ince it may inuence the pre-peak repone ignicantly. In fact, up to 50% of the pre-peak repone may be aociated with localization. Thi i due to the ductile behavior of the ECC which inuence the σ-w relationhip, making it very tough. Thi i illutrated in [6], where the oftening wa decribed uing a linear curve from peak tre to zero tre with w 2 = L f /2, where w 2 i the crack opening for which the tre tranfer i equal to zero, and L f i the ber length. The typical length of the ber in ECC i 8-12 mm. Thu, the preent model include the inuence of the σ-w relationhip, however, uing the bilinear approximation hown in Figure 1. Conideration of thi part of the material behavior will make it poible to determine the inuence of the localization on the FPBT repone. 3. The Hinge Model The modelling will follow the nonlinear hinge concept decribed in [7]. The hinge i illutrated in Figure 2a. The propagating crack i modelled within the element boundarie a a layer of independent pring element. Thee pring element are formed by incremental horizontal trip, and are attached at each end to a rigid boundary. Each boundary may rotate and tranlate uch that it may be joined with an uncracked beam modelled according to the claical beam theory. Figure 2b how the tre ditribution in the hinge when both cruhing and localization with tre free tail ha developed. Baed on thi ditribution, the entire hinge behavior from unet of loading to failure may be decribed. It prove convenient

4 y -2 y-1 2 y 0 y 1 y h N M y 0 localized crack d M N ½ h ½ h d y 2 y 3 y 4 h (a) (b) Figure 2: Geometry, loading and deformation of the hinge element (a) and tre ditribution in the hinge element when cruhing and localization of the crack ha developed (b) to introduce the mean value of the curvature and the ditribution of longitudinal train, κ and ε, repectively: κ = 2 ϕ ; ε = 2(y y 0 )κ (1) The deformation of an incremental trip i given by u(y) = ε (y) where i the width of the hinge. Thu, given the material parameter in Figure 1, the depth y 2, y 1, y 1 and y 2 may be determined: y 2 = y 0 ε c2 2ϕ, y 1 = y 0 ε c1 2ϕ, y 1 = y 0 + ε t1 2ϕ, y 2 = y 0 + ε t2 2ϕ (2) The total deformation, u(y), in the cracked tage may be determined a the um of the elatic deformation, the opening of the crack and the elatic unloading due to the tre drop caued by the localization: u(y) = ε t2 + w(y) f t2 σ w (y) = ε + w(y) + σ w(y) (3) E ul E ul Here, the crack opening in the cracked incremental trip in the depth y, w(y), ha been introduced. The aociated crack bridging tre, σ w (w(y)) i given by σ w (w(y)) = f t2 (b i a i w(y)), i = 1, 2 where b i = (1, b 2 ), ee Figure 1, and where a 1 and a 2 are the rt and econd lope of the normalized σ-w relationhip, repectively. The unloading i aumed elatic and the tine i denoted E ul while ε i the unrecovered train after complete unloading: ε = ε t2 f t2 E ul (4) Combining Equation 1 and 3 reult in an expreion for the crack tre at depth y: σ w (w(y)) = (2(y y 0 )ϕ ε w(y)) E ul (5)

5 Utilizing the bilinear σ-w relationhip aumption, an expreion for the crack opening, w(y), and the tre, σ(y), may be derived: w(y) = 2(y y 0)ϕ ε ζ i, σ(y) = ζ i 2(y y 0 )ϕβ i + εβ i E ul 1 β i 1 β i where the dimenionle parameter β i and ζ i are dened by: (6) β i = f t2a i, ζ i = f t2b i, i [1, 2] (7) E ul E ul The depth y 3 and y 4 may be determined from Equation 6 by olving w 1 = w(y 3 ) and w 2 = w(y 4 ): y i+2 = y ϕ (ζ i + ε + (1 β i )w i ), i [1, 2] (8) The complete tre ditribution may now be etablihed for any given rotation of the hinge, ϕ, and thu, the normal force, N and the moment, M may be derived. Thi i done by piecewie numerical integration in the range y [0, h] uing the phae change parameter y i, i [ 2, 1,.., 4]. The calculation proceed by incrementing the hinge rotation, ϕ, and ubequently determining rt the depth of the neutral axi and then the hinge moment M. 4. Implementation of the hinge model in the FPBT Figure 3a how the FPBT. The clear pan of the beam i L t, while the ditance from the end to the loading point i L c. The contant moment pan i denoted by L m. The deection at midpoint of the beam i meaured with reference to the loading point. Since the moment i contant in the middle part of the beam, thi part may be modelled with incremental hinge element, all undergoing the ame rotation. Prior to localization of the crack in the hinge, the hinge rotation and the midpoint deection will be governed by the initial elatic part and the train hardening part of the σ ε curve. Thee contribution are both non-local and may be termed the non-localized rotation, ϕ nl, and the non-localized deection, u nl. Thi i the ituation in the upper part of Figure 3a. However, after the peak moment, M p, i reached, a tructural localization will take place, i.e. the crack will continue to open at one or two location, while the crack in the remaining part of the beam will cloe. One poible mechanim in thi ituation i hown in the lower part of Figure 3a, while other may be contructed by uperpoition of imilar mechanim. After the tructural localization, the total rotation of the hinge and the midpoint deection will include a localized part, ϕ l and u l. Modelling of the unloading require that a certain unloading branch i dened. Figure 3b how the aumed unloading branch for the cloing hinge with a common focu point with the coordinate (ϕ fp, M fp ). The opening hinge will determine the moment capacity of the beam ince here the material continue to degrade. The midpoint deection, u nl, and the load, P, prior to the tructural localization are readily obtained a: u nl = 1 8 κl2 m = 1 4 ϕ nl L2 m, P = 4M L t L m (9) After the tructural localization ha occurred, the total hinge rotation i given by: ϕ tot = ϕ nl + ϕ l. For a given total hinge rotation and calculated moment, M(ϕ tot ), thee contribution may be determined a:

6 ( ) Mp M ϕ nl = ϕ p (ϕ p ϕ fp ) M p M fp, ϕ l = ϕ tot ϕ nl (10) where M p i the peak moment and ϕ p the correpoinding rotation. The localized deformation, u l, i determined from the actual localization mechanim. For the ituation hown in the lower part of Figure 3a, the reult i: u l (ϕ l ) = 1 2 (L a L c ) ϕ l for L a L b (11) The total deection in the cae where tructural localization ha taken place i readily obtained a the um of Equation 9 and 11. Note, that other localization mechanim may be modelled by uperpoing the mechanim in Figure 3 uing dierent localization point. L c nl L m u nl L c M M p M onet of material localization nl peak moment opening hinge l L t cloing hinge u l fp p tot L a l L b M fp (a) (b) Figure 3: Deformation of the FPBT during loading. Initially, no tructural localization i preent, while later, a tructural localization occur (a). Schematic moment-rotation curve for the non-linear hinge howing the incorporation of a imple elatic unloading criterion (b). 5. FEM calibration of the FPBT model The hinge model introduce the hinge width parameter,, which need to be determined through an FEM calibration. Invetigation on normal concrete and FRC applying the hinge model on the Three Point Bending Tet (TPBT) indicate that may be elected roughly a h/2 for all material choice and beam geometrie, ee e.g. [8]. However, for the FPBT it appear that the FEM reult are approximated mot preciely uing /h=0.9. Figure 4 how a comparion between the hinge model and the FEM reult for dierent beam height. The remaining parameter for the beam are: L t =500 mm, L m =250 mm and thickne t=100 mm. The material parameter have been elected a: f t1 =4 MPa, f t2 =6 MPa, E t1 =E c1 =22000 MPa, E t2 =100 MPa and a 1 = mm 1, where E c1, E t1 and E t2 denote the lope of the σ-ε relationhip. Compreive oftening and cruhing ha been diregarded in thi comparion by electing large value of f c1 and f c2. The σ-w relationhip ha been approximated with a linear curve by electing mall value of a 2 and b 2 while the

7 4 Normalized load P/Pel [-] material localization 100 onet of train hardening 50 h = 10 mm tructural localization Normalized deflection u/u el [-] Figure 4: FEM reult (thin line) compared with the hinge model reult (thick line) (a) for a FPBT for three dierent beam height with /h=0.9. The white circle mark the tranition point of the hinge model. calculation of a 1 i baed on the aumption that w 2 = L f /2, L f = 12 mm. The train capacity of the material i approximately 2% (ε t2 0.02). The tructural localization of the FEM model occur ymmetrically around the midection at both loading point (Figure 3a with L a = L c uperpoed with the ame ituation mirrored). Thi i due to a mall tre concentration at thoe point caued by the concentrated loading. The choice of in Figure 4 i made baed on an attempt to t the point of tructural localization for all beam height. The curve in Figure 4 are normalized with regard regard to the elatic deection, u el, and load, P el, which may be calculated uing Equation 2 and 9. The FEM and hinge model curve how a high degree of correlation. Prior to the material localization, where only the tenile train hardening account for the non-linearity of the load-deection repone, the reult are eentially identical. However, after the material localization i initiated, the hinge model reult become dependent on, but with the optimal choice of, the dierence are till limited. 6. Invere analyi Method for the extraction of the train hardening propertie of ECC i the ubject of only one paper in the literature [4]. However, thi paper doe not include the eect of localization of the crack. Uing the hinge model decribed in the previou ection, localization i included, and a precie reult i obtained by imply uing the quared um of dierence between the experimental reult and the model reult a object function: min (f t1,f t2,e t1,e t2 ) ( ) 2 N ˆPi P i (12) i=1 in which ˆP i i the load obtained by the hinge model for a certain deection, u i, while P i P i

8 denote the experimentally obtained load for the ame deection. The data point on the graph mut be equiditantly paced on the curve in order to give each part the ame weight in the calculation. Note that the compreive propertie are not included in the optimization ince they may be determined initially by a imple compreion tet. Alo knowledge of the σ-w relationhip i unimportant - in fact it turn out, in comparion with FEM reult that the train hardening propertie may be determined without knowing thi relationhip. Table 1: Reult from invere analyi with dierent beam height, h [mm], together with an analyi on the enitivity of the election of a 1 [mm 1 ] and [mm]. The train hardening material parameter were elected to f t1 = 4 MP a, f t2 = 6 MP a, E t1 = 22 GP a and E t2 = 100 MP a. The number in the table repreent the error on the determination of the material parameter in percent h a E t1 1.2 % +1.1 % +2.3 % +4.4 % % +2.3 % +2.3 % +2.3 % E t % 0.8 % +1.0 % +2.1 % +8.0 % 0.6 % 1.4 % 4.0 % f t1 1.1 % 0.1 % 0.5 % 0.9 % 4.2 % 0.4 % 0.3 % 0.1 % f t2 1.7 % 0.2 % 0.8 % 1.1 % 1.9 % +1.4 % +2.4 % +5.4 % Thi i hown in Table 1 where variou FEM reult have been ubject to the method of invere analyi. The correlation between the reult i remarkably good. If moderate h/l m value are elected the invere analyi i very precie - only extremely thin beam, h = 10 mm, or very thick beam, h = 100 mm give le ne reult. Thi i particular een for the thick beam where E t1 i determined with an error of 30%. Note that the election of i unimportant - mall a well a high value of give good reult, which how that a cloe t of the point of tructural localization i not important. It i alo intereting to notice that the train hardening propertie can be determined regardle of the fact that a part of the total deection of the beam i caued by crack localization. However, note that thi i only true if the material localization point i predicted accurately. Finally, a demontrated in Table 1, even if a 1 i elected a 32 time larger than the value ued in the FEM model, accurate reult are obtained. 7. Experimental reult The method for extraction of the σ-ε relationhip of ECC ha been teted on an ECC mix with the mix proportion given in Table 2. In total 9 beam were cat with varying beam height, h = 40, 50 and 60 mm. The remaining beam dimenion were L m = 250 mm, t = 100 mm and L t = 500 mm. The cating direction wa perpendicular to the beam height in order to avoid any material variation throughout the beam height. The beam were cat in wooden water-tight mold and tored in the concrete lab for the rt 24 hour, covered with platic. Then, they were water cured until the teting began at an age of 7 day. The FPBT etup wa deigned uch that both upport and loading point could rotate independently around two axe: a horizontal axi parallel to the beam and a horizontal axi perpendicular to the beam. Furthermore, each loading point could move vertically independently of each other. Thi ytem enure that no redundant force were acting on the beam.

9 Table 2: Mix deign. The cement type wa CEM I FA i y ah, while SP i uper platicizer Mix Cement Sand Quart FA (milled) FA Water SP Fiber kg/m The reult from the experiment are hown in Figure 5a. A expected, there i ome catter, which uually i the cae for ECC. Figure 5b how a comparion between one experimental curve and the correponding output from the invere analyi. The correlation i convincing and it demontrate the ability of the model to capture real ECC behavior h = 60 mm h = 50 mm Load [kn] h = 40 mm Load [kn] Deflection [mm] Deflection [mm] (a) (b) Figure 5: Experimental reult (a), and a comparion between one experimental curve and the outcome of the invere analyi (b), (h = 40 mm, /h = 0.9, unloading according to Figure 3a with L a = L c + 20 mm) The average output from the dierent beam height are hown in Table 3. It i evident that all beam height give the ame reult, which i rather promiing for the method ince thi exclude the poibility of tructural inuence on the material parameter determined. 8. Concluion Thi paper preent a new method for interpretation of the load-deection repone of the ECC FPBT etup, by incorporating cruhing, the bilinear compreive and tenile σ-ε relationhip and the bilinear σ-w relationhip of the ECC material. The method prove very accurate when compared with FEM reult. It ha been demontrated that localization of the crack occur prior to the peak load, and thu, only a part of the total deection of an ECC beam may be attributed to the train hardening repone.

10 Table 3: Experimental reult obtained through the invere analyi. The reult for each beam height repreent the average of three experiment. The latter column give the average and tandard deviation for all nine beam. h [mm] Average Std. dev. E t1 [GPa] E t2 [GPa] f t1 [MPa] f t2 [MPa] ε t1 [µε] ε t2 [%] However, a method propoed for invere analyi prove able to ditinguih between the train hardening and the crack localization part of the deection, thu enabling extraction of the train hardening propertie. Compared with FEM, thi method give very precie reult. The method alo perform well in comparion with experiment. However, complementary UTT experiment are needed in order to nally validate the model. 9. Acknowledgement The Knud Højgaard Foundation i gratefully acknowledged for upporting the work, while Kuraray Europe i credited for the generou delivery of the PVA ber. Prof. V. C. Li and Aoc. Prof. H. Stang are alo highly recognized for inpiring thi work. 10. Reference [1] Li, V. C On Eng. Cemen. Comp. (ECC). J. Adv. Conc. Tech., 1(3), [2] Li, V. C., Wu, C., Wang, S., Ogawa, A. & Saito, T Interf. Tailoring for Strain- Hardn. Polyvinyl Alc. Eng. Cemen. Comp. (PVA-ECC) J. ACI Mat., 99(5), [3] Stang, H. & Li, V. C Claication of Fiber Reinf. Cem. Mat. for Struc. Appl. BEFIB, RILEM, [4] Kanakubo, T., Shimizu, K., Katagiri, M., Kandra, T., Fukuyama, H. & Rokugo, K. Eval. of Tenile Prop. for DFRCC - Reult of Round Robin Tet by JCI Techn. Committee Proc. JCI Symp. on DFRCC, (in Japanee) [5] Maalej, M. & Li, V. C Flexural/Tenile Strength Ratio in Eng. Cemen. Comp. ASCE J. Mat. Civ. Eng., 6(4), [6] Kabele, P. & Horii, H Anal. Model for Frac. Behavior of Peudo Strain-Hardn. Cemen. Comp. Concr. Libr. Int., 29, [7] Oleen, J. F Fictitiou Crack Prop. in Fiber-Reinf. Conc. Beam. J. Eng. Mech., 127(3), [8] Øtergaard, L Early-Age Frac. Mech. and Crac. of Conc. Dept. of Civil Eng., Tech. Univ. of Denmark