Analysis of Steel Fiber-Reinforced Concrete Elements Subjected to Shear

Transcription

1 ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title No. 113-S25 Analyi o Steel Fiber-Reinorced Conete Element Subjected to Shear by Seong-Cheol Lee, Jae-Yeol Cho, and Frank J. Vecchio In thi paper, a rational analyi procedure i preented or modeling the hear behavior o teel iber-reinorced conete (SFRC) element. In the development o the analyi procedure, the Diturbed Stre Field Model (DSFM), baed on the Modiied Compreion Field Theory (MCFT), i modiied by implementing contitutive model or SFRC, which are derived rom the Divere Embedment Model (DEM). For the contribution o teel iber, a local tine matrix or iber ha been developed eparately rom thoe or conete matrix and conventional reinorcement. The compoite element tine matrix or an SFRC element with conventional reinorcement i then derived by uperpoing the three local tine matrixe. In the element tine matrix, the eect o hear lip at a ack i alo taken into account by conidering the reitance due to teel iber againt hear tre on ack urace. Through comparion with the tet reult o SFRC panel previouly reported in the literature, it i hown that the actual hear behavior o SFRC panel are accurately predicted by the propoed analyi procedure, not only or the hear trength but alo or the hear train at the ailure. Through implementation into inite element analyi program, the analyi procedure developed in thi paper can be ueul in the modeling o SFRC member and tructure alo containing conventional reinorcement. Keyword: inite element; hear train; hear trength; teel iber; teel iber-reinorced conete (SFRC). INTRODUCTION It i well known that teel iber-reinorced conete (SFRC) exhibit ductile tenile behavior even ater acking becaue o the pullout behavior o teel iber bridging the ack. For the pat everal decade, much reearch ha been conducted to evaluate the tenile behavior o SFRC through experimental program 1-5 or analytical model development In CEB-FIP Model Code 2010, 11 a the reult o thee invetigation, the tenile reitance o SFRC i included in the deign o conete tructure; the tenile tre o SFRC can be evaluated rom the reult o ourpoint bending tet on notched beam in accordance with BS EN Several reearcher ocued on whether teel iber can be ued to replace hear reinorcement in a reinorced conete beam; Dinh et al. 13 meaured the ultimate hear trength o SFRC beam with conventional reinorcement while Suetyo et al. 14 experimentally invetigated the hear behavior o SFRC panel with conventional reinorcement. ACI conditionally allow the ue o teel iber o 0.75% in volume a minimum hear reinorcement in reinorced conete beam. Although remarkable progre ha been made in SFRC reearch, a deibed previouly, SFRC i not yet widely ued in tructural application becaue it i till diicult to theoretically predict the tructural behavior o SFRC ACI Structural Journal/March-April 2016 member with conventional reinorcement. Evidence o thi wa given by Suetyo et al., 16 who analyzed the SFRC panel they teted by implementing the Variable Engagement Model (VEM) 7 and direct tenion tet reult within a nonlinear inite element analyi method baed on the Diturbed Stre Field Model (DSFM). 17,18 They ound that the analyi reult were igniicantly aected by the election o tenion tiening/otening model, hear lip on ack urace, and ack pacing parameter. Thi indicate the neceity to develop a more rational analyi procedure to predict the tructural behavior o SFRC member with conventional reinorcement, epecially in regard to the hear behavior, a many reearch eort 13,14,16 have ocued on SFRC to partially or entirely replace the hear reinorcement in conete member. In thi paper, thereore, an advanced analyi procedure or the hear behavior o SFRC panel with conventional reinorcement i preented. For the development o the analyi procedure, tree and train in the panel are explored in depth through a theoretical approach, and then recently developed model baed on the Divere Embedment Model (DEM) 8,9 are employed to evaluate the tructural behavior o SFRC with or without conventional reinorcement. RESEARCH SIGNIFICANCE Recently, everal contitutive model, 8-10,19 baed on the DEM, were developed to repreent the tructural behavior o SFRC with or without conventional reinorcement. Although the uniaxial behavior o SFRC member can be modeled reaonably well with them, the hear behavior o SFRC member wa not rationally captured. The analyi procedure developed in thi paper make it poible to predict not only the uniaxial behavior but alo the hear behavior o SFRC member with conventional reinorcement. The procedure deibed can be eaily implemented in both an exiting ectional analyi program 20 and inite element analyi program 21 : each layer or each element can be modeled a a reinorced conete element ubjected to biaxial tre condition and then it can be analyzed by employing the developed analyi procedure. Thu, the analyi procedure deibed will be ueul in modeling the tructural behavior o SFRC member and tructure with conventional reinorcement. ACI Structural Journal, V. 113, No. 2, March-April MS No. S R1, doi: / , wa received June 12, 2015, and reviewed under Intitute publication policie. Copyright 2016, American Conete Intitute. All right reerved, including the making o copie unle permiion i obtained rom the copyright proprietor. Pertinent dicuion including author cloure, i any, will be publihed ten month rom thi journal date i the dicuion i received within our month o the paper print publication. 275

2 Fig. 1 Local tenile tre ditribution in SFRC element. STRESSES AND STRAINS IN SFRC PLANAR ELEMENTS Stree in SFRC element To analyze the hear behavior o an SFRC planar element, one mut evaluate the tree in three material: namely, the conete matrix, conventional deormed reinorcing bar, and teel iber. Figure 1 how the tre ditribution in each material in the principal tenile direction. A illutrated in the igure, with no conideration given to tenion otening eect, the tre in the conete matrix i zero at a ack, while the tree in the other two material are highet at a ack. Between the ack, becaue tenile tree are hared by the conete matrix due to the bond mechanim between conete matrix and the other two material, the tre contribution rom the three material vary along the principal tenile direction. For implicity o calculation, and a already adopted and veriied in the DSFM and the MCFT, 22 ue o the average tree concept, rather than a rigorou conideration o tre variation, wa expected to work well. Thereore, the tree in a SFRC element were idealized by employing a meared ack model, a illutrated in Fig. 2. For the tenile tree reited by the conete matrix and conventional reinorcement, the average tree can be obtained rom tenion-tiening and tenion-otening model and bilinear or trilinear model, repectively. On the other hand, the tenile tre in teel iber at a ack can be evaluated rom the DEM or the impliied DEM (SDEM). 10 In thi paper, thereore, the coeicient α avg ha been introduced to relate the tenile tre in teel iber at a ack with the average tenile tre. Detail will be preented in the dicuion o contitutive relation that ollow. In addition to the average tree, the hear tre and lip on ack urace hould be conidered becaue they could be igniicant actor inluencing the hear behavior o SFRC panel ubjected to hear. A preented in Fig. 3, the hear Fig. 2 Average tenile tree in SFRC element with meared ack. Fig. 3 Average and local tree in SFRC element: (a) average tree; and (b) local tree. tre on a ack urace can be evaluated through a comparion between the average tree under meared ack tatu and the local tree at a ack, a in the ollowing equation v ρ ( )inθ co θ ( 1 α ) in θ (1) ci,, i, i, i ni, ni, avg i where v ci, i the hear tre on ack urace that i reited by the conete matrix, including aggregate interlock. In Eq. (1), the term (1 α avg ) inθ repreent the contribution o teel iber bridging a ack. Without teel iber, the equation converge to the equation or normal reinorced conete panel a preented in the DSFM. From the hear tre on the ack urace, hear lip on the ack urace can be calculated rom the conventional model dicued in the Contitutive relation ection that ollow. Strain in SFRC element In the DSFM ormulation, total train and net train are eparately deined conidering the eect o lip; the 276 ACI Structural Journal/March-April 2016

3 total train are the apparent (meaured) train that include deormation due to lip at a ack, while the net train are conete train deduced rom the apparent train by ubtracting the train due to the hear lip at a ack, a illutrated in Fig. 4. The train due to the hear lip at a ack are calculated a ε ε ε γ x y x ( γ / 2)in( 2θc) ( γ / 2)in( 2θc) γ co( 2θ c ) (2) where γ δ / i an average hear lip train; and θ c i an inclination o principal net tenile train in conete. Hence, the relationhip between the total train, [ε], and the net train, [ε c ], are a ollow [ε] [ε c ] + [ε ] (3) where [ε] [ε x ε y γ xy ] T ; and [ε c ] [ε cx ε cy γ cxy ] T. For the evaluation o tree within the three material (conete matrix, teel iber, and conventional reinorcement) in a SFRC planar element, the total train are compatible with the contitutive law or conventional reinorcement while the net train are compatible with the contitutive law or conete matrix and teel iber. CONSTITUTIVE RELATIONS Model or SFRC without conventional reinorcement SFRC can exhibit a ductile compreive behavior ater peak tre that i quite dierent rom that o normal conete becaue the teel iber mitigate againt pot-peak plitting ack. In addition, the train at the peak tre o SFRC i generally greater than that o normal conete. 23 To relect the eect o teel iber on the compreive behavior, the model propoed by Lee et al. 23 ha been adopted, which had been developed rom the tet reult o 48 cylinder pecimen with 150 mm (6 in.) diameter and 300 mm (12 in.)height. c2 c2max A( εc2 / εc ) B A 1+ ( εc2 / εc ) where A B 1/[1 ( c /ε c E c )] or the pre-peak acending branch, and A (V l /d ) 0.957, B ( c /50) [ (V l /d ) ] A or the pot-peak decending branch, and c2max i the maximum compreive tre conidering the compreion otening eect, a ollow 21 c2max ( 028. ). ε / ε c c1 c2 where ε c1 /ε c2 > The elatic modulu o SFRC and the train at the compreive trength, i not known, can be etimated rom Eq. (6) and (7) (4) (5) Fig. 4 Strain in SFRC element: (a) SFRC element deormation; (b) net train or SFRC; (c) train due to lip; and (d) total train. E V l c c d V l c d ε c To repreent the tenile behavior o SFRC which exhibit ductile behavior even ater acking, the ormulation o the DEM 8,9 and the SDEM 10 have been adopted. In the DEM, the tenile tre at a ack due to teel iber can be rigorouly calculated or a given ack width, through a double numerical integration cheme, to evaluate the average iber tenile tre at a ack a ollow (6) (7) α V σ,,avg (8) 2 l 2 π 2 where σ, avg, σ, ( la, θ) in θdθdl 0 a. 0 l Becaue the double numerical integration make the calculation quite demanding, the DEM wa impliied to the SDEM in which the bond mechanim or the pullout behavior o teel iber and mechanical anchorage eect due to end-hook are eparately evaluated, a ollow t + eh (9) ACI Structural Journal/March-April

4 It hould be noted that Eq. (14) i applicable to both iber tenion model: DEM and SDEM. Conequently, the average tenile tre due to teel iber over the average ack pacing can be calculated by multiplying the reult o Eq. (8) or (9) with the coeicient α avg. In addition to the tenile tre carried by the teel iber, the tenile tre taken by the conete matrix hould alo be taken into account to evaluate the total tenile tre utained by the SFRC. For thi, the ollowing tenion otening model 7 can be employed. ct e cw (15) Fig. 5 Ditribution o tenile tre in teel iber at ditance rom a ack. (Note: 1 mm in.) where K eh K t l w α V K τ,max d l t t 2 li 2w α V K τ,max d eh eh eh β w 3 β 1 + w 3 w ( ) or w or w < w w eh β eh eh or w < eh βeh 2( w eh eh ) l li or eh w < wc r l li 2 2 li 2w l li li 2li l Keh, i or w < 2 2 (10) (11) (12) (13) It hould be noted that the SDEM i applicable in mot cae but only the DEM i uitable or SFRC in which iber rupture i expected; iber rupture i not conidered in the SDEM. A illutrated in Fig. 1, the tenile tre due to teel iber varie between ack, o the average or the tenile tre due to teel iber through ack hould be evaluated. Employing the DEM, it i poible to evaluate the magnitude o the tenile tre tranmitted rom the teel iber to the conete matrix between ack, a illutrated in Fig. 5. Baed on the reult o the DEM, a imple model or the coeicient α avg over the average ack pacing ha been derived a α avg 2 l 10. (14) 55. where the coeicient c i 15 and 30 or conete and mortar, repectively. Conequently, the tenile tre o SFRC without conventional reinorcement can be evaluated by uperpoing the contribution o teel iber and conete matrix, a ollow SFRC + ct (16) Model or SFRC with conventional reinorcement In SFRC member with conventional reinorcement, the local ineae in tenile tre within a reinorcing bar at a ack i tranmitted back into the conete matrix between ack; thi mechanim i commonly reerred to a the tenion-tiening eect. Thereore, the average tenile tre o conete matrix due to the bond mechanim between the conete matrix and the conventional reinorcement hould be evaluated. For the average tenile tre due to the tenion tiening eect, the ollowing model 19 i employed. cts, cm εc 1 (17) where the coeicient c conider the eect o teel iber, evaluated a c (1/0.034)(l /d )[(100V ) 1.5 /M 0.8 ] or end-hooked iber. The bond parameter M i calculated rom M A c /( d b π), in millimeter. For reinorced conete member without teel iber, the above equation converge to the conventional tenion tiening model 17,18,24 a c become 0.6. It i noted that the tenile tre due to the tenion tiening eect hould not be le than the tenile tre due to the tenion otening eect. 17,18 Conequently, the tenile tre in the conete matrix can be evaluated by conidering both the tenion-tiening eect and the tree tranmitted by teel iber, a ollow c1 c,ts + (1 α avg ) coθ (18) A local yielding o conventional reinorcement at a ack mut alo be conidered, the upper limit o the tenile tre in a conete matrix can be deined a ollow 2 c 1 ρi(, i i, )co θni, + ( 1 αavg) coθ (19) i The tenile tree carried by teel iber and by tenion otening are both calculated or a given ack width, while 278 ACI Structural Journal/March-April 2016

5 the tenile tre due to the tenion tiening eect i calculated or a given average tenile train. Thu, the average ack pacing i required to etablih a relationhip between the ack width and the average tenile train. In thi paper, the average ack pacing model recently derived by Deluce et al. 25 rom the tet reult with 47 R/FRC member ubjected to uniaxial tenion ha been employed. That i b kk 2 ca + k mi (20) where c 1.5a gg ; k 1 0.4; k ; k 3 1 [min(v, 0.015)/0.015][1 (1/k )]; a gg i the maximum aggregate ize, in millimeter, and mi b 4 ρ i π d 1 i, 2 bi, co 4 θ i (21) ρi, α V co 2 θi + k (22) i d d bi, Conequently, the average ack width can be imply evaluated by multiplying the average tenile train and the average ack pacing w ε c1. In SFRC member containing conventional reinorcement, multiple ack occur with variable ack width even under uniormly ditributed tre condition. Becaue the tenile tre tranmitted by teel iber i alo inluenced by ack, it i neceary to check whether the tenile tre reited by teel iber calculated or the average ack width i not greater than that or the maximum ack width. To conider the maximum ack width, the ollowing model 25 ha been adopted in thi paper. w Vl,. + max w d (23) Model or hear lip at ack in SFRC with conventional reinorcement In SFRC panel with conventional reinorcement, the eective direction o the tenile tre taken by teel iber ao a ack may deviate rom the principal tenile tre axi within the conete becaue o hear lip at a ack, a preented in Fig. 3. Thereore, it i important to conider the eect o teel iber on the hear lip at a ack. Becaue hear tre at a ack reited by iber bridging a ack ha been already ubtracted in Eq. (1), by employing the hear tre-lip model or normal reinorced conete propoed by Vecchio and Lai, 26 the hear lip at a ack can be evaluated or the hear tre at the ack calculated rom Eq. (1). Thereore, the hear lip at a ack can be calculated accordingly, a ollow δ δ 2 Ψ 1 Ψ (24) 05. v + v cmax co δ w w 020. cc ( ) (25) where Ψ v ci, /v c,max ; v cmax c / ( 24w / a + gg 16), in MPa; v co cc /30; and cc i the conete cube trength, in MPa. Conequently, the direction o the tenile tre due to teel iber i deviated by θ, which i calculated rom the ollowing relationhip θ δ tan 1 (26) Becaue θ i combined with only the tenile tre due to teel iber,, all the equation in thi paper converge to the equation or normal reinorced conete member i no teel iber are provided. Model or conventional reinorcement To repreent the average tenile tre-train relationhip o conventional reinorcement, a trilinear model can be employed a the ollowing equation w ε E or ε ε y (27) y or ε y < ε ε h (28) y + (ε ε h )E h or ε > ε h (29) FINITE ELEMENT IMPLEMENTATION Once the principal train are calculated and the appropriate contitutive law are applied or principal compreive and tenile repone, the analyi o a acked SFRC panel proceed a or an orthotropic element in the manner deibed by Vecchio. 18 From the principal tree and train, the local material tine matrix or conete can be evaluated by employing ecant moduli, a illutrated in Fig. 6. E c1 0 0 [ Dc ] 0 Ec Gc (30) ( ) where E c1 c1 /ε c1 ; E c2 c2 /ε c2 ; and Gc Ec 1 Ec2 Ec 1+ Ec2. In the ame manner a with the conete matrix, the local material tine matrixe or teel iber and each conventional reinorcement can be deined a ollow where E i,i /ε,i. ρiei 0 0 [ D ] i or the i-th conventional reinorcement (31) ACI Structural Journal/March-April

6 D E or teel iber (32) where E 1 α avg /ε c ; and ε c (ε c1 + ε c2 )/2 + [(ε c1 ε c2 )/2] co2θ. Ater tranorming the local material tine matrice to the global axi, the global tine matrix or a SFRC panel i obtained a ollow [ D] [ Dc]+ D + D i, (33) where [D c ] [T c ] T [D c ] [T c ]; [D ] [T ] T [D ] [T ]; [D,i ] [T,i ] T [D,i ] [T,i ]; and 2 2 co ψ in ψ coψin ψ 2 2 [ T ] in ψ co ψ coψin ψ 2coψin ψ 2coψ in ψ (co 2 ψ in 2 ψ) i (34) Fig. 6 Contitutive relation and ecant moduli: (a) conete matrix under compreion; (b) conete matrix under tenion; (c) conventional reinorcement; and (d) teel iber. where ψ θ c or the conete (that i, inclination o the net principal tenile tre); ψ θ c + θ or teel iber (that i, inclination o the tenile tre due to teel iber; reer to Fig. 3); and ψ α i or i-th conventional reinorcement (that i, orientation o reinorcing bar). With the global tine matrix or a SFRC element etablihed, the relationhip between the tree induced by external load and the total train can be expreed a ollow [σ] [D][ε] [σ 0 ] (35) where [σ] [σ x σ y τ xy ] T, [ε] [ε x ε y γ xy ] T, and [σ 0 ] [D c ][ε ]. COMPARISON WITH SFRC PANEL TEST RESULTS Tet panel detail For veriication o the propoed analyi procedure, eight SFRC panel teted by Suetyo et al. 14 were analyzed. The tet variable, ummarized in Table 1, were conete compreive trength (with target trength o 50 and 80 MPa [7.25 and ki] or C1 and C2, repectively), iber volumetric ratio (0.5, 1.0, and 1.5% or V1, V2, and V3, repectively), and iber type (end-hooked iber: F1, F2, and F3). The SFRC panel were 890 x 890 mm quare and 70 mm thick (35 x 35 x 2.75 in.), and contained longitudinal teel reinorcement in the ratio o 3.31% with a yield trength o MPa (80.1 ki); no tranvere reinorcement wa provided. The panel were loaded in a pure hear tre condition, uing a monotonically ineaing orcecontrolled protocol, until the panel ailed. Material propertie or conete and teel iber are given in Table 1. Detail o the SFRC panel and a repreentative ailure mode are preented in Fig. 7. Comparion o analyi reult and tet reult In Table 2 and Fig. 8, the hear trength and train at ailure predicted by the propoed analyi procedure are preented and compared with the tet reult. A evident in the table and igure, the analyi reult are in good agreement with the tet; or the eight pecimen, the ratio o the calculated hear trength to meaured hear trength had a mean o 0.99 and a coeicient o variation (CoV) o 0.11; or the hear train at peak tre, the ratio had a mean o 0.99 and a CoV o Figure 9 and 10 provide comparion o the calculated and experimentally oberved repone or all panel pec- Table 1 Conete and teel iber material propertie o SFRC panel 14 Conete Steel iber Specimen c, MPa (ki) l, mm (in.) d, mm (in.) σ u, mm (in.) V, % C1F1V (7.45) 50.0 (1.97) 0.62 (0.024) 1050 (152) 0.5 C1F1V (7.75) 50.0 (1.97) 0.62 (0.024) 1050 (152) 1.0 C1F1V (7.21) 50.0 (1.97) 0.62 (0.024) 1050 (152) 1.5 C1F2V (8.66) 30.0 (1.18) 0.38 (0.015) 2300 (334) 1.5 C1F3V (6.60) 35.0 (1.38) 0.55 (0.022) 1100 (160) 1.5 C2F1V (11.43) 50.0 (1.97) 0.62 (0.024) 1050 (152) 1.5 C2F2V (11.10) 30.0 (1.18) 0.38 (0.015) 2300 (334) 1.5 C2F3V (8.99) 35.0 (1.38) 0.55 (0.022) 1100 (160) ACI Structural Journal/March-April 2016

7 Table 2 Comparion o analyi and tet reult or SFRC hear panel Analyi Tet Analyi/Tet Specimen τ xy,u, MPa (ki) γ xy,u, 10 3 τ xy,u, MPa (ki) γ xy,u, 10 3 ττ xy, u, Analyi C1F1V (0.466) (0.512) C1F1V (0.744) (0.750) C1F1V (0.968) (0.779) C1F2V (0.892) (0.969) C1F3V (0.689) (0.811) C2F1V (0.959) (1.001) C2F2V (0.987) (0.915) C2F3V (0.791) (0.808) Average CoV xy, u, Tet γ xy, u, Analyi γ xy, u, Tet Fig. 8 Comparion o tet and analyi reult or hear trength and hear train at ailure: (a) hear trength; and (b) hear train at ailure. Fig. 7 Detail o SFRC panel: (a) reinorcement coniguration; and (b) ailure mode o C1F1V3. imen. A een in Fig. 9, the hear tre-train repone o the SFRC panel were predicted well by the propoed analyi procedure. The minor deviation oberved are mainly due to variance in the prediction o the SFRC tenion model (that i, DEM), a revealed in Fig. 10. The ACI Structural Journal/March-April

8 Fig. 9 Comparion o tet and analyi reult or hear tre-train repone. total tenile tre in SFRC (that i, the um o tenile tree due to teel iber and conete matrix) wa overetimated or C1F1V3, wherea it wa underetimated or C1F3V3. The hear at ailure or C2F1V3 wa underetimated becaue o rupture o the teel iber, which i more diicult to capture analytically. Panel C1F1V1 exhibited a relatively brittle behavior ater acking becaue o it low iber volumetric ratio; or thi panel, there i ome dierence in the peak train repone, although the predicted trength accurately matche with the tet reult. Panel C1F3V3 wa overetimated becaue it exhibited principal tenile tre higher than C2F3V3; or thi panel, ome tet anomalie may have occurred. Overall, rom Fig. 9 and 10, it can be inerred that any diepancy in the hear behavior o SFRC panel between the prediction and the tet reult i mainly due to error in the evaluation o tenile tre provided by teel iber, and i not ymptomatic o the developed analyi procedure. In pite o thee diepancie, the prediction or the hear behavior o SFRC panel are generally within an acceptable range. In general, the hear tre-train repone were well predicted when the uniaxial tenile behavior o SFRC wa well predicted. Conequently, it can be concluded that the actual tructural behavior o SFRC panel ubjected to pure hear i well predicted by the propoed analyi procedure, not only or the hear tre-train repone but alo principal tenile tre-train repone in the panel. 282 ACI Structural Journal/March-April 2016

9 Fig. 10 Comparion o tet and analyi reult or SFRC principal tenile tre-train repone. CONCLUSIONS In thi paper, an advanced analyi procedure ha been developed or modeling the hear behavior o SFRC element. Dierent rom the approach generally adopted in previou work in the literature, where the tenile tre o conete wa imply ineaed to conider the contribution o teel iber, the contribution o the three material conete matrix, conventional reinorcement, and teel iber were conidered eparately in the propoed analyi procedure. For the evaluation o the tree due to teel iber, the contitutive model or the tenile behavior o SFRC (DEM and SDEM) and acking behavior model or SFRC member with conventional reinorcement (tenion tiening model and ack pacing model) were employed o that the contribution o teel iber wa rigorouly taken into the account. The propoed analyi procedure wa veriied through comparion with the tet reult o SFRC panel. With the propoed analyi procedure, not only the hear trength but alo the hear train at ailure were predicted uiciently well. In addition, it wa hown that the propoed analyi procedure capture well the principal tenile tre-train repone o SFRC, which i a dominant behavior in SFRC panel with conventional reinorcement. Conequently, it i concluded that the actual hear behavior o SFRC panel can be modeled reaonably accurately by ACI Structural Journal/March-April

10 the propoed analyi procedure. Becaue it can be eaily implemented in ectional analyi program or inite element analyi program that are baed on a meared rotating ack approach, the propoed analyi procedure will be ueul or prediction o the actual SFRC member or tructure with conventional reinorcement. AUTHOR BIOS ACI member Seong-Cheol Lee i an Aociate Proeor at KEPCO International Nuclear Graduate School (KINGS), South Korea. He received hi PhD rom Seoul National Univerity, Seoul, Korea, in 2007, and wa then a potdoctoral reearcher in the Department o Civil Engineering at the Univerity o Toronto, Toronto, ON, Canada. Hi reearch interet include the hear behavior o conete tructure and the analyi o pretreed conete tructure and iber-reinorced conete member. ACI member Jae-Yeol Cho i an Aociate Proeor in the Department o Civil & Environmental Engineering at Seoul National Univerity, where he received hi PhD. Hi reearch interet include nonlinear analyi and optimized deign o reinorced and pretreed conete tructure, material modeling, and imilitude law or dynamic tet o conete tructure. Frank J. Vecchio, FACI, i a Proeor in the Department o Civil Engineering at the Univerity o Toronto. He i a member o Joint ACI-ASCE Committee 441, Reinorced Conete Column, and 447, Finite Element Analyi o Reinorced Conete Structure. Hi reearch interet include nonlinear analyi and deign o reinorced conete tructure, contitutive modeling, perormance aement and orenic invetigation, and repair and rehabilitation o tructure. ACKNOWLEDGMENTS Thi reearch wa upported by the Korea Intitute o Energy Technology Evaluation and Planning (KETEP), unded by the Minitry o Knowledge Economy (No. 2011T ), upported by the Nuclear Saety Reearch Program through the Korea Foundation o Nuclear Saety (KOFONS) granted inancial reource rom the Nuclear Saety and Security Commiion (NSSC) (No ), and upported by the Intitute o Contruction and Environmental Engineering at Seoul National Univerity. Support wa alo provided by the Natural Science and Engineering Reearch Council o Canada (NSERC). NOTATION [D] element tine matrix or SFRC with conventional reinorcement [D c ] material tine matrix or conete [D ] material tine matrix or teel iber [D ] i material tine matrix or i-th conventional reinorcement d b diameter o conventional reinorcement d iber diameter E c elatic modulu o conete E elatic modulu o conventional reinorcement E h train hardening modulu o conventional reinorcement c compreive trength o conete cylinder c,ts average tenile tre in conete due to tenion tiening eect c1 principal tenile tre in conete c2 principal compreive tre in conete c2max peak compreive tre in conete conidering compreion otening eect acking trength o conete ct tenile tre in conete due to tenion otening eect eh tenile tre due to mechanical anchorage eect o end-hooked teel iber tenile tre at ack due to teel iber SFRC tenile tre o SFRC average tre in conventional reinorcement local tre (at ack) in conventional reinorcement t tenile tre due to rictional bond behavior o teel iber y yield trength o conventional reinorcement l iber length l i ditance between mechanical anchorage or end-hooked iber average ack pacing in principal tenile tre direction in conete eh lip at maximum tenile orce due to mechanical anchorage o iber with inclination angle o 0 degree to normal o ack urace lip at rictional bond trength or iber with inclination angle o 0 degree [T] rotation tranormation matrix V iber volumetric ratio v ci, hear tre on ack urace w average ack width w,max maximum ack width α avg coeicient to relate tenile tre at a ack due to teel iber with average tenile tre α iber orientation actor β, β eh coeicient to conider eect o iber lip on longer embedment ide on rictional behavior and mechanical anchorage eect, repectively δ lip diplacement along ack urace [ε] apparent (total) average train in element including ack lip train [ε c ] average train in conete ε c train at conete compreive trength ε c1 average train in conete in principal tenile tre direction ε c2 average train in conete in principal compreive tre direction [ε ] equivalent average train due to dicontinuou lip along ack ε h hardening train o conventional reinorcement ε y yield train o conventional reinorcement γ hear train due to lip along ack urace θ c inclination o principal tenile tre in conete θ angle between tenile tre direction due to teel iber and principal tenile tre direction in conete θ n angle between conventional reinorcement and normal to ack ρ reinorcement ratio [σ] total tre o SFRC element with conventional reinorcement [σ 0 ] oet tre due to ack lip σ,,avg average iber tenile tre at ack σ, (l a, θ) iber tenile tre at ack or given horter embedment length l a and iber inclination angle θ τ eh,max equivalent bond trength due to mechanical anchorage o teel iber rictional bond trength o teel iber τ,max REFERENCES 1. 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