# post-cracking modelling of concrete beams reinforced with synthetic

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post-cracking modelling of concrete beams reinforced with synthetic

6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy POST-CRACKING MODELLING OF CONCRETE BEAMS REINFORCED WITH SYNTHETIC FIBERS Byung Hwan Oh, Ji Cheol Kim, Young Cheol Choi,Dae Gun Park Dept. of Civil Engineering, Seoul National University, Korea Abstract The post-cracking behavior of newly-developed structural synthetic fiber reinforced concrete beams is analyzed. The pullout behavior of single fiber is identified by tests and employed in the model to predict the post-cracking behavior. A probabilistic approach is used to calculate the effective number of fibers across crack faces and to calculate the probability of non-pullout failure of fibers. The proposed theory is compared with test data and shows good correlation. The proposed theory can be efficiently used to predict the load-deflection behavior, moment-curvature relation, load-crack opening width relation of synthetic fiber reinforced concrete beams. 1. Introduction Most important application of fibers is to prevent or control the tensile cracking occurring in concrete structures [1-11]. Recently, structurally-efficient synthetic fibers have been developed by authors and coworkers [9]. These synthetic fibers have advantages compared to steel or other fibers in that they are corrosion-resistant and exhibit high energy-absorption capacity. The purpose of the present study is to explore experimentally and theoretically the cracking resistance and post-cracking behavior of newly-developed structural synthetic fiber reinforced concrete beams. 2. Models for post-cracking behavior 2.1 Concept of analysis A fiber reinforced concrete beam as shown in Fig. 1 has been considered for the analysis of post-cracking behavior. Fig. 1 shows the failure mode of a beam with the crack mouth opening displacement. Fig. 2 shows the strain and stress distributions for cracked section of fiber reinforced concrete (FRC) beam. In order to obtain the post-cracking behavior of FRC beams, the stress-strain relations of concrete in compression and tension, and the 1045 6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy stress-crack width relation after cracking must be properly defined. 2.2 Stress-strain relation of concrete in compression The following commonly-used compressive stress-strain equations were employed. Hc H ( c ) 2 ] for 0 d H c d H 0 H0 Ho fc f c ' [2 fc f c ' [1 (1) 0.15 (H c H 0 )] for H 0 d H c d 0.003 0.004 H 0 (2) in which fcː = compressive strength of concrete, H0 = the strain at peak stress. The force C in compression zone can be written as follows. C Ef c ' bc (3) where Eҏ= the factor for average stress, b = width of beam, and c = the depth of neutral axis from the top face of a beam. The value of E can be obtained by equating the area under actual stress-strain curve to the area under rectangular stress block as follows[10] H cf ³ f dH H cf ³ c Df c ' H cf ; Then E f c dH c (4) f c ' H cf 0 P/2 c 0 P/2 L L/3 L/3 h=L/3 '0 P/2 P/2 c dT dG dT h yi wi CMOD Vc f1 2T f2 f3 f4 fi fN dT c c' force axial strain of top fiber(Hx) displacement x=L Hcf Vc = Vc(Hc) stress x=0 'c H0 strain L/3 fi = f(wi , Di , li) dCMOD dT dT Fig. 2 Schematic view of forces and stresses acting on cracked section of FRC beam The location of the compression force, Jc, from the top fiber can be obtained as Fig. 1 Failure mode of FRC beam 1046 6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy H cf ³H 1 J f dH c c c 0 (5) H cf H cf ³ f dH c c 0 2.3 Stress-strain relation of concrete in tension 2.3.1 Post-cracking behavior of concrete after tensile strength The cracking starts to occur right after the tensile stress reaches the tensile strength and the tensile stress decreases as the crack width increases. The strain at first cracking, Hcr, can be obtained as follows.G fr (6) Ect in which fr = tensile strength of concrete, Ect = elastic modulus before cracking. The stresses after cracking depend on the widths of cracks. It is reasonably written here based on the Gopalotratnam & Shah’s model [5]. H cr V ct e fc kwO (7) in which Vct = the tensile stresses after cracking, k = empirical constant = 60.8, w = crack width(in mm), O= empirical constant = 1.01. 2.4 Calculation of sectional forces and deflections of FRC beams The deformation of concrete beams without reinforcing bars is usually localized at central position as shown in Fig. 1. The cracked portion at central location acts as a plastic hinge. The displacement 'n at compression face can be obtained from the compressive strain distribution Hx (see Fig. 1).G '0 L ³ H x dx 0 H cf 2 L (8) The slope(rotation angle) of beam, dT, may also be obtained from Fig. 1.G '0 (9) 2c The deflection at central position is obtained from the slope of the beam as follows. dT | dG dT L 2 (10) The crack mouth opening displacement (CMOD) at the bottom surface is also written as dCMOD 2[dT (h c)] (11) 1047 6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy The internal resisting moment, Me, of the beam can be derived from the stress distributions of concrete in compression and tension zones and also the pullout forces of all the fibers across the crack plane as shown in Fig. 2. c N (12) ³ V c (bd y ) ¦ fi 0 i 1 0 N ³ V c bdy y ¦ ( fi yi ) 0 i 1 c Me (13) The pullout force fi of the fiber of ith layer in Equation 13 should be obtained from the relation of bond stress and bond slip of a fiber and they directly depend on the crack widths of the beam(see Equation 14). ª CMOD º (14) « ( h c ) » yi ¼ ¬ The fibers are randomly distributed at the crack plane and this effect must be considered appropriately to calculate the fiber forces. Soroushian and Lee[11] proposed the number of fibers per unit area, N1, as follows. Vf N1 D (15) Af wi where, Dҏ= fiber orientation factor, Vf = fiber content, and Af = cross-sectional area of fibers. If the boundary of the structure restrains the arbitrary orientation of the fibers, the orientation factor for a specified direction becomes larger. Soroushian and Lee[11] reported the orientation factor for two-side-restrained case as follows. h/2 ³ E1dy D1 df /2 for h l f h/2 (16) lf /2 lf D1 ³ E1dy df /2 hl f / 2 (17) 0.405(1 l f / h) S /2 r where E1 ³ ³l f cos T cos\dTd\ 0 r0 l f (S / 2)r with r0 sin 1 ( d f / l f ) and r sin 1 (2 y / l f ) (18) The tensile force(Fi) resisted by fibers at each layer of tensile zone of the beam can be obtained by multiplying the fiber force fi of that layer by the number of fibers. The force fi of single fiber can be derived from the bond stress-slip relation, which is also 1048 6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy dependent upon the crack width wi of ith layer(see Fig. 2). This bond stress-slip relation of structural synthetic fiber will be directly obtained by tests, which will be described in the following section. The development length Lt of a fiber required for not to be pulled out at the crack plane may be derived as shown in Equation 19. Lt fi (19) 60W u in which 60 = perimeter of a fiber, and Wu = bond strength of a fiber. Therefore, the actual embedment length of fibers should be larger than the required development length in order not to be pulled out. Fig. 3 summarizes the fiber forces, required anchorage lengths, and the probabilities of non-anchorage failure for various layers of cracked section of a beam. The probability of non-anchorage(or non-pullout) failure Pr can be derived as follows by two cases. (1) Case 1 : Required anchorage length d the half of fiber length(Lf d lf/2)G This is the case that the half of the actual fiber length is larger than the required anchorage length. This can be written in a formalized equation as follows. Pr 1 (h c) Lt / 2 (h c)l f /2 L 1 l f (20) f (2) Case 2 : Required anchorage length > the half of fiber length(Lf > lf/2) This is the case that the required anchorage length is larger than the half of actual fiber length. This can be expressed as the following equation. (l Pr / 2)k / 2 f (h c)l / 2 f l /2 1 f 2 Lt l 1 f 4 Lt (21) in which k l f (h c) / Lt . 2 Fig. 3 Fiber force, required anchorage length and probability of non-anchorage failure Finally, the tensile forces resisted by fibers can be obtained from the number of fibers at each layer, effective orientation factor D, and the probability of non-anchorage failure Pr. 1049 6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy 3. Tests for post-cracking behavior of synthetic FRC beams 3.1 Pullout test for single fiber The authors developed recently new structural synthetic fibers which are of crimped type with the length of 50mm. Fig. 4 shows the photograph for actual pullout test arrangement. Fig. 5 depicts the average pullout load versus slip relation obtained from the present tests. The bond load-slip equation obtained from the tests may have the following form. Fp aS (22) b cS d where a, b, c, and d are the constants to be obtained from test data, and Fp= pullout load(kN), S = slip in mm unit. Fig. 4 Photograph of pullout test Fig. 5 Load-slip relation for crimped-type synthetic fiber 3.2 Flexural tests for structural synthetic fiber reinforced concrete beams The concrete beams reinforced with structural synthetic fibers have been tested to obtain the flexural behavior including the load-deflection behavior, load-CMOD relations, and moment-curvature relations. The water-cement ratio was 0.45 and the fiber content was 1 percent of total concrete volume. The dimension of the beam was 100u100u400mm and the span length was 300mm. The load was applied in third point loading in displacement control manner with the rate specified in the static testing standard.G 4. Analysis of test results and comparisons with theory Fig. 6(a) shows the load-deflection curves obtained from the present tests for the beam with structural synthetic fiber volume of 1 percent. Fig. 6(b) also compares the test data with the theory proposed in the previous section. It can be seen that the theoretical predictions fairly well agree with test data even after post-cracking ranges. The salient feature of the post-cracking behavior of structural synthetic fiber reinforced concrete beams is that the resisting load drops down right after first cracking, probably 1050 6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy due to initial slip of fibers at crack plane, and then starts to increase due to structurally effective synthetic fibers in tensile region. Fig. 6(b) also exhibits the similar behavior of structural synthetic fiber reinforced concrete beam for fiber volume of 1.5%. It is seen again in Fig. 6(b) that the proposed theory agrees very well with the measured data, even up to the large deflection of the beam. 20 20 Experimental (Vf = 1.5%) Experimental (Vf = 1.0%) 16 Analytical (Vf = 1.0%) Load (kN) Load (kN) 16 12 8 4 Analytical (Vf = 1.5%) 12 8 4 0 0 0 0.5 1 1.5 2 0 2.5 0.5 1 1.5 2 2.5 Midpoint Deflection (mm) Midpoint Deflection (mm) (a) fiber content: Vf = 1.0% by volume (b) fiber content: Vf = 1.5% by volume Fig. 6 Load-deflection curve for synthetic FRC beams Fig. 7 describes the load-CMOD relations for two different fiber volume contents. It is noted here that, at the same loads after cracking, the FRC beam with larger volume of fibers exhibits much smaller CMOD values. This is indeed a great beneficial effect of structural synthetic fibers. Fig. 8 shows the relation between CMOD and central deflection of FRC beam. These relations are almost similar for different fiber volume contents. Therefore, this relation of CMOD versus central displacement may be regarded as a material property for structural synthetic fiber reinforced concrete beam. 5 20 Vf = 1.0% Vf = 1.0% 4 Vf = 1.5% CMOD (mm) Load (kN) 16 12 8 Vf = 1.5% 3 2 1 4 0 0 0 0.5 1 1.5 2 0 2.5 0.5 1 1.5 2 2.5 Midpoint Deflection (mm) CMOD (mm) Fig. 7 Effect of fiber content on loadCMOD relation Fig. 8 Relation between CMOD and deflection 1051 6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004 20-22 September 2004, Varenna, Italy 5. Conclusions A realistic model for post-cracking behavior of structural synthetic fiber reinforced concrete beams is developed in this study. The randomness of orientation of fibers and the effective number of fibers at the crack plane were first considered. New concept of the probability of non-pullout failure of fibers at the crack plane was then introduced and derived in this study. The pullout tests for fibers were conducted and an appropriate relation between bond forces and slips was derived in order to calculate the pullout forces of structural synthetic fibers at the crack plane. The load-CMOD relation, CMODdeflection, and moment-curvature relation were also reasonably predicted by the proposed theory. The present study also indicates that the relation between CMODs and central displacements is almost same for the beams with different fiber volumes and, therefore, this relation can be regarded as a material property for structural synthetic fiber reinforced concrete members. The present study allows more realistic analysis and application of recently-developed structural synthetic fiber reinforced concrete beams. References 1. ACI Committee 544, ‘Design Considerations for Steel Fiber Reinforced Concrete’. 2. ASTM C 1018-89, ‘Standard Test Method for Flexural Toughness and First-Crack Strength of Fiber Reinforced Concrete(Using Beam with Third-Point Loading)’, Book of ASTM Standards, V. 04.02, ASTM, Philadelphia: (1991) 507-513. 3. Banthia, N. & Trottier, J., ‘Concrete Reinforced with Deformed Steel Fibers, Part I : Bond-Slip Mechanism’, ACI Material Journal, 91 (5) (1994) 435-446. 4. Ezeldin, A.S. & Balaguru, PN., ‘Normal and High Strength Fiber Reinforced Concrete under Compression’, Journal of Materials in Civil Engineering, ASCE, 4(4) (1992) 415-427 5. Gopalaratnam, V.S. & Shah, S.P., ‘Softening Response of Plain Concrete in Direct Tension’, ACI Journal, Proceeding 82 (3) (1985) 310-323 6. Leung, C., Geng, Y.P., ‘Micromechanical Modeling of Softening Behavior In Steel Fiber Reinforced Cementitious Composites’, Int. J. Solids Structures, 35 (31-32) (1998) 4205-4222. 7. Li, V.C., et al., ‘Influence of Fiber Bridging on Structural Size-effect’, Int. J. Solids Structures, 35 (1998) 4223-4238 8. Oh, B.H., ‘Flexural analysis of reinforced concrete beams containing steel fibers’, Journal of Structural Engineering, ASCE, 118(10) (1992) 2691-8. 9. Oh, B.H., ‘Cracking and flexural behavior of structural synthetic fiber reinforced concrete beams’, KCI Journal, 14 (6) (2002) 900-909 10. Park, R. & Paulay. T., ‘Reinforced concrete structures’, John Wiley and Sons, Inc., New York, N.Y. (1975) 11. Soroushian, P. & Lee, C., ‘Distribution and Orientation of Fibers in Steel Fiber Reinforced Concrete’, ACI Material Journal, 87 (5) (1990) 433-439 1052