Mechanical Properties and Modeling of Amorphous Metallic Fiber-Reinforced Concrete in Compression

International Journal of Concrete Structures and Materials, Jun 2016

The aim of this paper is to investigate the compressive behavior and characteristics of amorphous metallic fiber-reinforced concrete (AMFRC). Compressive tests were carried out for two primary parameters: fiber volume fractions (V f ) of 0, 0.3, 0.6 and 0.8 %; and design compressive strengths of 27, 35, and 50 MPa at the age of 28 days. Test results indicated that the addition of amorphous metallic fibers in concrete mixture enhances the toughness, strain corresponding to peak stress, and Poisson’s ratio at high stress level, while the compressive strength at the 28-th day is less affected and the modulus of elasticity is reduced. Based on the experimental results, prediction equations were proposed for the modulus of elasticity and strain at peak stress as functions of fiber volume fraction and concrete compressive strength. In addition, an analytical model representing the entire stress–strain relationship of AMFRC in compression was proposed and validated with test results for each concrete mix. The comparison showed that the proposed modeling approach can properly simulate the entire stress–strain relationship of AMFRC as well as the primary mechanical properties in compression including the modulus of elasticity and strain at peak stress.

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Mechanical Properties and Modeling of Amorphous Metallic Fiber-Reinforced Concrete in Compression

International Journal of Concrete Structures and Materials Vol.10 1976-0485 Mechanical Properties and Modeling of Amorphous Metallic Fiber- Reinforced Concrete in Compression Ngoc-Hieu Dinh 0 Kyoung-Kyu Choi 0 Hee-Seung Kim 0 0 Soongsil University , Seoul , Korea The aim of this paper is to investigate the compressive behavior and characteristics of amorphous metallic fiberreinforced concrete (AMFRC). Compressive tests were carried out for two primary parameters: fiber volume fractions (Vf) of 0, 0.3, 0.6 and 0.8 %; and design compressive strengths of 27, 35, and 50 MPa at the age of 28 days. Test results indicated that the addition of amorphous metallic fibers in concrete mixture enhances the toughness, strain corresponding to peak stress, and Poisson's ratio at high stress level, while the compressive strength at the 28-th day is less affected and the modulus of elasticity is reduced. Based on the experimental results, prediction equations were proposed for the modulus of elasticity and strain at peak stress as functions of fiber volume fraction and concrete compressive strength. In addition, an analytical model representing the entire stress-strain relationship of AMFRC in compression was proposed and validated with test results for each concrete mix. The comparison showed that the proposed modeling approach can properly simulate the entire stress-strain relationship of AMFRC as well as the primary mechanical properties in compression including the modulus of elasticity and strain at peak stress. compressive strength; amorphous metallic fibers; fiber-reinforced concrete; Poisson's ratio; strain at peak stress; modulus of elasticity; stress-strain curve 1. Introduction For several decades, significant research has been performed to improve binder materials (cement, fly ash, and slag) and adhesives (superplasticizer, etc.) suitable for high performance construction materials (Constantinides et al. 2003; Jeong et al. 2015; Le et al. 2014; Sriravindrarajah et al. 2012; Divsholi et al. 2014; Sorensen et al. 2014) . Based on the progress in concrete technology, high strength concrete (HSC) with compressive strength of 100 MPa or above has been developed and applied in various construction fields. Nonetheless, such HSC generally shows severe brittle behavior in compression compared to normal strength concrete (NSC) (Gettu et al. 1990; Wittmann 2002) ; HSC suddenly loses its load carrying capacity after reaching peak compressive strength. Partly due to the brittlenesspoffifffiffi HSC, ACI 318-14 (2014) limits the maximum value of fc0 to be 8.3 MPa in shear design of concrete structures. In addition, the ACI Manual of Concrete Practice (1998) limit the use of HSC in earthquake resisting structures. For this reason, considerable research has been performed to improve the characteristics and the ductility of concrete (in particular HSC) in compression by the addition of fibers in concrete. In a study by Ezeldin and Balaguru (1992) , the use of fibers in both NSC and HSC ranging from 35 to 85 MPa was found to increase toughness, compressive strength, strain corresponding to peak stress, and the secant modulus of elasticity. In addition, Ou et al. (2011) investigated the compressive stress–strain behavior of steel fiberreinforced concrete (SFRC) with a high reinforcing index, which was defined as the product of fiber volume fraction and fiber aspect ratio. Their study indicated that the addition of hooked-end fibers to concrete matrix increases both material toughness and strain at the peak stress, while the modulus of elasticity and compressive strength of SFRC are less affected. Moreover, Chi et al. (2012) performed experiments to investigate the uniaxial compressive behavior of steel-polypropylene hybrid fiber-reinforced concrete (HFRC). It was observed that the compressive behavior of concrete can be improved by the addition of hybrid fibers and showed a variation according to fiber volume fractions of steel and polypropylene fibers and aspect ratio of steel fibers. Based on the test results, the optimum proportion of hybrid fibers was also investigated to achieve enhanced compressive strength and ductile compressive behavior. Recently, Srikar et al. (2016) investigated the residual compressive properties of polypropylene fiber- reinforced concrete exposed to high temperature by using digital image correlation method. It was reported that the polypropylene fibers improved the post-peak residual compressive strength and the toughness of concrete subjected to temperatures up to 300 C. The stress–strain curves also show a significantly improvement in post peak behavior with increasing fiber volume fraction at all level of exposure temperatures. Recently, newly-developed fibers, known as amorphous metallic fibers (AMF), have been investigated by numerous researchers. Choi and Ku (2014) performed flexural tests on amorphous metallic fiber-reinforced concrete (AMFRC) and it was found that the addition of the fibers in concrete could significantly enhance flexural strength as well as toughness; a higher fiber volume fraction of up to 0.75 % could result in a more enhanced flexural behavior of concrete. Moreover, Hameed et al. (2010) and Won et al. (2012) demonstrated that the presence of amorphous metallic fibers increased the flexural toughness and strength and was more effective than the conventional steel fibers in terms of flexural toughness and strength. Additionally, Choi et al. (2014) investigated the shrinkage and corrosion resistance of amorphous metallic fiber-reinforce cement composites and conclude that amorphous metallic fibers displayed higher corrosion resistance than did steel in every degradation environment, and plastic shrinkage crack control performance was excellent. More recently, experiments performed by Choi et al. (2015) showed that the addition of AMF into concrete could significantly reduce the free drying shrinkage by 24 %; and in the restraint shrinkage tests (ring tests and slab tests), AMF delay crack development time as well as reduced crack width to 36.5–74.6 % according to the fiber volume fraction. Notwithstanding this, the mechanical properties of AMFRC in compression were not adequately investigated in the above studies. In this current study, a compressive test on AMFRC was performed with variations of fiber volume fraction and concrete compressive strength. Based on the test results, primary compressive characteristics were investigated in terms of the modulus of elasticity, Poisson’s ratio, strain corresponding to peak stress, and the toughness. In addition, an analytical model was proposed for simulating the entire compressive stress–strain curves of AMFRC as functions of the fiber volume fraction and concrete compressive strength. 2. Experimental Program 2.1 Materials In order to investigate the influence of amorphous metallic fibers on the compressive behavior of concrete, three different concrete mix proportions were employed with the design compressive strengths of 27, 35 and 50 MPa at the 28-th day. The detailed proportions are presented in Table 1. The concrete matrix consisted of Type I Portland cement, natural river sand having a fineness modulus of 2.65, and crushed limestone with the maximum aggregate size of 13 mm and fineness modulus of 6.40. The grading of fine aggregate and coarse aggregate conformed to the requirement specified in ASTM C33 (2013). A super-plasticizer was used as an admixture with a dosage of cement content of 0.2–0.8 % to achieve adequate workability of fresh concrete in the presence of metallic fibers. The fibers used in this test are a special type, called amorphous metallic fibers (AMFs). Figure 1 illustrates the configuration of AMFs. They are made of an amorphous alloy of the Fe family with a density of 7200 kg/m3, straight ribbon type of 1.6 mm width, 30 mm length, 0.029 mm thickness (equivalent aspect ratio L/df = 125), modulus of elastic of 140 GPa, and yield tensile strength of 1700 MPa. Moreover, as shown in the scanning electron microscopy (SEM) image (Fig. 2), the amorphous metallic fibers also have a rough surface, which has been found to improve the bond strength between the fibers and the cementitious matrix (Choi et al. 2015) . In this study, four different fiber volume fractions were employed: 0 % (control type), 0.3, 0.6, and 0.8 %. The dosages corresponding to the fiber volume fractions are 0, 21.6, 43.2, and 57.6 kg/m3, respectively. 2.2 Mixing, Casting, and Curing Procedure The concrete was mixed using a compulsive concrete mixer with a capacity of 0.6 m3. First, the dry aggregate and cement were mixed for 1 min. Then, 80 % of water was added and mixing was continued for a further 1 min. The mixing was then continued with the fibers for 2 min. Finally, the remaining water along with the superplasticizer was poured gradually and mixing was continued for an additional 3 min. The preparation procedure of specimens in this study follows ASTM C 192 (2015); the fresh concrete mix was cast in standard cylinders of 100 mm diameter and 200 mm height in three equal layers with each layer being rodded with 25 strokes using a tamping rod. For each batch, at least 5 specimens were cast and then covered with a wet cloth to prevent evaporation. After 24 h, they were demoulded and cured according to ASTM C31 (2015) in lime-saturated water at a temperature of 23 ± 1.7 C before the loading test. As shown in Fig. 3, the water-heater was used to maintain the temperature in the container at 23 ± 1.7 C due to the low temperature of the laboratory in winter. 2.3 Test Setup The compressive test for cylindrical specimens was carried out according to ASTM C39 (2015). Before testing, the cylinders were flattened at both ends to ensure smooth surfaces to transfer the uniform load. The test was conducted using a hydraulically-powered machine with a capacity of 1000 kN. A data acquisition system was connected to the computer to record the load and strain data. In this test, similar to a previous study (Xie et al. 2015) , a compressometer was used with two high sensitive linear variable displacement transducers (LVDTs) on two opposite locations of the specimen as shown in Fig. 4. The strain was then calculated as the ratio between the average displacement of two LVDTs and the gauge length of 100 mm. To measure the transverse strain of the specimens, two concrete strain gauges with a length of 60 mm were attached horizontally at the mid height of the cylinder using a quickcuring adhesive, as illustrated in Fig. 5. In order to attach the strain gauges to the uneven and rough surfaces of the concrete specimens, the voids on the surfaces were filled with a suitable pre-coating material. To obtain the entire stress– Design strength (MPa) 27 strain curve, a compression test was carried out using a displacement control method with a velocity of 0.005 mm/s according to a previous study (Chi et al. 2012) . Both the displacement and strain data were collected using the data acquisition system. The test was intentionally ended at a vertical deformation of about 5 mm. 60mm concrete strain gauge LVDTs high performance superplasticizer or optimum concrete mix proportion with the presence of fly ash and silica fume. 3.2 Characteristics of Hardened Concrete The important parameters characterizing the compressive behavior of plain and amorphous metal fiber-reinforced concrete were investigated and are summarized in the form of normalization in Table 2: compressive strength, the strain at peak stress, ultimate strain, elastic modulus, and toughness index. 3.2.1 Compressive Strength and Strain at Peak Stress The strength test results are summarized in Table 2. In this table, the compressive strength of each test specimen 3. Experimental Results and Discussions 3.1 Properties of Fresh Concrete The characteristics of fresh concrete using amorphous metal fibers were investigated based on an air content test and slump-flow test for each concrete mixture according to standards of ASTM C231 (2014) and ASTM C143 (2015). The air content test results for different concrete mixes are given in Fig. 6. They indicate that the presence of amorphous metallic fibers in fresh concrete mix lead to a slight increase of air void content in the range of 0.4–0.6 %. Figure 7 illustrates the relationship between fiber volume fraction and slump value according to different grades of concrete mix. It shows that a higher volume fraction of amorphous metallic fibers resulted in lower workability. In particular, for the fiber volume fraction of 0.8 % of 50 MPa concrete mix, the slump value obtained from experiment is 100 mm, which decreased by 47 % compared to plain fresh concrete mix (190 mm). To address this problem, further research is needed aiming to enhance the slump flow of fresh amorphous metal fiber-reinforced concrete mixture, by using 27 MPa Mix 35 MPa Mix 50 MPa Mix 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Fiber volume fraction, Vf (%) Plain 27 normalized by the average value for each concrete mix are reported. The measured average compressive strengths of plain concrete at the 28-th day were 28.69, 39.27, and 52.30 MPa, while the design strengths were 27, 35, and 50 MPa. Figure 8 presents the typical stress–strain curve of AMFRC with the design strength of 27 MPa. In general, amorphous metallic fibers do not affect the compressive strength of concrete. The difference in compressive strength between the amorphous metal fiber-reinforced concrete and plain concrete is within 0.17–2.93 % (see Table 2), which is negligible. According to previous studies (ACI Manual of Concrete Practice 1998; Neves and Gonlalves 2000) , the fibers transfer stress across the micro cracks even in the compressive test, which could lead to higher compressive strength. However, the fibers could cause some disturbance Strain at peak stress Ultimate strain (Defined at 50% compressive strength) in the concrete matrix, which results in more voids in the concrete micro structure (Neves and Gonlalves 2000) and has a negative effect on compressive strength. Due to the trade-off between micro-crack bridging and additional voids, the influence of amorphous metal fibers on compressive strength is not significant. In Figs. 8 and 9, the compressive strain, eof, corresponding to the peak stress of AMFRC is presented. eof shows an increasing trend with increasing fiber volume fraction, V , f and this trend is consistent for all specimens, regardless of concrete compressive strength. For example, at 0.8 % fiber volume fraction, eof is almost 0.00291–0.00362, which is almost 130 % that (0.0022–0.00286) of plain concrete. Figures 8 and 10 present the ultimate compressive strain, eou, corresponding to the stress at which concrete is crushed. The value of eou is computed at limited stress ru = 0.5 fc in the descending branch of the stress–strain curve according to the CEB-FIP Model Code (1990 ). It also shows an increase of ultimate strain accompanying the increase of fiber volume fraction, Vf. This phenomenon is consistent even in the case of higher compressive strength specimens of 50 MPa, which is normally considered to be brittle. This demonstrates that the addition of amorphous metallic fibers assists in efficiently controlling the propagation of micro-cracks into macro-cracks during the post-peak stage of concrete. For HSC with the compressive strength of 70–100 MPa, additional future tests need to be conducted to verify the effect of amorphous metallic fibers on the ultimate compressive strain of HSC. 3.2.2 Failure Modes The typical failure modes of AMFRC specimens observed under uniaxial compressive test in this study are illustrated in Fig. 11 (Cunha et al. 2006) . Vertical cracks were formed immediately before reaching peak stress, and inclined shear failure cracks were formed at the beginning of the softening phase. This process lasted until each specimen had completely failed. The vertical cracks and the crack opening can be explained by fracture mode I, while the inclined shear crack causing crack opening and sliding can be explained by the combination of fracture modes I and II (Cunha et al. 2006) . Figure 12 illustrates the typical crack patterns of 50 MPaspecimens for different fiber volume fractions. It is evident that the presence of fibers modified the crack patterns of the concrete cylindrical specimens, whereby the plain specimen was governed by a few macro cracks. However, the AMFRC specimen showed many vertical or diagonal cracks with a 0.2 0.4 0.6 0.8 Fiber volume fraction, Vf (%) (a) 27 MPa series 0.2 0.4 0.6 0.8 Fiber volume fraction, Vf (%) (b) 35 MPa series 1 0.004 f o , ss 0.003 e r t s k ea 0.002 p t a ian 0.001 r t S 0.004 f o ,s 0.003 s e r t s ak 0.002 e p t a in 0.001 a r t S 0 0 0 0 1 1 0.012 0.2 0.4 0.6 0.8 Fiber volume fraction, Vf (%) (b) 35 MPa series 1 Fig. 10 Variation of compressive ultimate strain, eou, of AMFRC for different fiber volume fractions and concrete compressive strengths. (a) (b) (c) (b) (c) σ σ Separation σ τ Mode I τ Mode I + II Sliding w σ Fig. 11 Failure mechanism of AMFRC under compression (Cunha et al. 2006) . (a) (b) (c) (d) Fig. 12 Crack patterns of specimens with different fiber volume fractions (Vf) after failure: a Vf = 0 %; b Vf = 0.3 %; c Vf = 0.6 %; and d Vf = 0.8 %. narrow crack width, and in particular, the specimen with 0.8 % fiber volume fraction did not show significant concrete spalling and macro cracks even after failure. This phenomenon is partly due to the bridging mechanism of fibers during the post-cracking stage of specimens (ACI Manual of Concrete Practice 1998) . 3.2.3 Modulus of Elasticity The value of elastic modulus of concrete in compression was obtained from the stress–strain curve of each specimen based on the method according to ASTM C469 (2014): Ec ¼ ðS2 S1Þ=ðe2 0:00005Þ ð1Þ where Ec = modulus of elasticity (MPa), S2 = stress corresponding to 40 % of ultimate load (MPa), S1 = stress corresponding to a longitudinal strain, e1, of 0.00005 (MPa), and e2 = longitudinal strain produced by stress S2. Figure 13 presented the influence of fiber volume fraction on modulus of elasticity in the case of concrete compressive strength of 35 MPa. In addition, the calculated results for each specimen in the form of normalization and the mean value for the different concrete mix are reported in Table 2. Based on this, the correlation between the fiber volume fraction and elastic modulus for various compressive strengths is presented in Fig. 14. Similar to a previous study on steel fiber-reinforced concrete (Ou et al. 2011) , the presence of amorphous fibers in the concrete matrix also causes a decrease in the modulus of elasticity. In particular, for the fiber volume fraction of 0.8 %, the elastic modulus plastic material. In this study, the toughness is measured using a toughness index, which is defined as the ratio of area under the stress–strain curve at a strain of 0.015 and the area at peak stress, as show in Fig. 15. The value of 0.015 has also been used in Nataraja et al. (1999) and Bhargava et al. (2006) . Based on the above definition using the strain of 0.015, in the present study, the toughness index obtained from each specimen and the average value are presented in Table 2. Figure 16 shows the influences of amorphous fiber volume fraction on the toughness index according to the concrete compressive strength. In plain concrete, the value of the toughness index varies between 2.12 and 2.43 for the three different compressive strengths. In the 0.8 % fiber volume fraction, the toughness index reaches almost 4.5, which is about two times that of plain concrete. It is f'c ) a P M ( O A B o Area OACD Toughness index, I = Area OAB Strain C D 0.015 Prediction by Eq.(3) was reduced by 24.03, 17.09, 19.47 % in the case of 27, 35 and 50 MPa concrete, respectively. This is because the fibers parallel to the load direction act as voids in the concrete matrix (Rossi and Harrouche 1990; Mansur et al. 1999) . 3.2.4 Toughness Index Toughness is one of the important parameters explaining the ductility of the compressive behavior of concrete after reaching peak stress. The toughness is defined as the area under the stress–strain curve. To quantify the effect of fiber on the toughness of concrete in compression, Fanella and Naaman (1985) proposed the toughness index (TI) while Ezeldin and Balaguru (1992) proposed another approach using the toughness ratio (TR), which is the ratio of toughness of fiber-reinforced concrete to that of a rigid 45 40 35 )aP 30 (M 25 s rse 20 tS 15 10 5 0 0 40 )a 35 P (G30 c E, 25 s luu 20 d om15 tisc 10 laE 5 0 40 )a 35 (PG30 Ec 25 , lsu 20 u od 15 icm10 t lsa 5 E 0 0 0 Vf =0% Vf =0.3% Vf =0.6% Vf =0.8% 0.001 0.002 Axial strain (mm/mm) 0.2 0.4 0.6 0.8 Fiber volume fraction, Vf (%) (c) 50 MPa series Fig. 13 Typical stress–strain relationship (35 MPa series). Fig. 15 Definition of compressive toughness index (I). 0.2 0.4 0.6 0.8 Fiber volume fraction, Vf (%) (a) 27 MPa series 0.2 0.4 0.6 0.8 Fiber volume fraction, Vf (%) (b) 35 MPa series 1 1 1 40 )aP 35 I 5 , edx 4 n i ss 3 e n gh 2 u o T 1 1.2 fc 1 / c ,ss0.8 e r t sd0.6 e z i la0.4 m r o N0.2 0 0 Vf = 0.8% Vf = 0.6% Vf = 0.3% Vf = 0% 0.2 0.4 0.6 0.8 Fiber volume fraction, Vf (%) interesting that the toughness index also shows almost the same increasing trend even in higher strength concrete of 50 MPa, which is usually expected to be brittle. This proves that the presence of amorphous metallic fibers in concrete enhances the post failure behavior in compression. 3.2.5 Poisson’s Ratio In terms of Poisson’s ratio ( ), which is the ratio of transverse to axial strain, Fig. 17 shows the typical variation in Poisson’s ratio according to the normalized stress of different concrete compressive strengths. It can be observed that in case of 27 MPa, below the normalized stress level of 40 %, Poisson’s ratio shows almost constant value of 0.2; however, after stress level of 60 %, the Poisson’s ratio of the plain specimen tends to increase suddenly, leading to a stretch in the curve. In the case of 35 and 50 MPa concrete, Poisson’s ratio shows almost constant value of 0.2 below 60 % stress level and a sharp increase after 80 % stress level for plain specimen. This result was expected because the external layer is damaged more quickly than the core of the specimen (Ferretti et al. 1999) . On the other hand, in the case of amorphous fiber-reinforced specimens, the increased fiber 0.006 content in the concrete matrix induces less stretch in the normalized stress—Poisson’s ratio curves. Interestingly, in the case of the fiber volume fraction of 0.8 %, the Poisson’s ratio of 0.2 is maintained as constant until the peak stress level. This phenomenon can be observed for all cases of 0.8 % fiber volume fraction of different the concrete compressive strengths (27, 35, and 50 MPa). It was demonstrated that the presence of amorphous metal fibers in a concrete matrix has limited expansion in the horizontal direction, resulting in less destructive external surface and contributing to the stabilization of Poisson’s ratio at high stress level. 4. Modeling and its Validation of Compressive Characteristics of AMFRC 4.1 Modulus of Elasticity Mansur et al. (1999) proposed an analytical equation to evaluate the elastic modulus considering the fiber volume fraction and compressive strength as pffiffiffi Ef ¼ ðA B Vf Þ 3 fc ð2Þ where Ec is in GPa, Vf is in percentage, and fc is in MPa. In this study, based on the regression analysis, the elastic modulus, Ecf , of amorphous metallic fiber-reinforced concrete was defined as a function of the elastic modulus, Ec, of plain concrete and fiber volume fraction, Vf ð%Þ Ecf ¼ Ecð1:04 0:3Vf Þ For the limited test data obtained in this study, the best fitting equation for Ec was defined as pffiffiffi Ec ¼ 4480 fc ðMPaÞ Figure 14 compares the prediction obtained from Eq. (3) and the test results. The figure shows a relatively good correlation between the experimental data and the predicted results. However, since the value of the elastic modulus of concrete strongly depends on the nature of coarse aggregate (Baalbaki et al. 1991; A¨ıtcin and Mehta 1990; Gutierrez and Canovas 1995) , for different aggregate types, the above equation needs to be adjusted. 4.2 Strain at Peak Stress In the test results observed in this study, it was found that the strain, eof , at the peak stress of the amorphous metallic fiber-reinforced concrete was strongly related to the strain at the peak stress of plain concrete and fiber volume fraction. Based on regression analyses for each group of tested specimens, an analytical equation to evaluate the strain at peak stress was proposed as eof ¼ eoð0:9 þ 0:46 Vf Þ where Vf = fiber volume fraction (%) and eo = strain corresponding to peak stress of plain concrete, determined from the regression analysis of the experiment data. For the ð3Þ ð4Þ ð5Þ 1.2 1 f c f / c σ, 0.8 s s e r ts 0.6 d e z i la 0.4 m r o N 0.2 0 1.2 1 f c f / σ,c 0.8 s s e trs 0.6 d e liz 0.4 a m r oN 0.2 0 limited test data obtained in this study, the best fitting equation for eo was defined as eo ¼ 6:05fc 0:667 Ec In Eq. (6), Ec is the modulus of elasticity of plain concrete Eq. (4). ð6Þ The predicted values of the compressive strain at peak stress of plain concrete and amorphous metal fiber-reinforced concrete from Eqs. (5) and (6) are compared with the experimental data in Fig. 18. Generally, the model shows a good prediction of the strain at peak stress according to compressive strength and fiber volume fraction. relationship for both plain concrete and AMFRC. The model was also used in Ezeldin and Balaguru (1992) , Mansur et al. (1999) , Nataraja et al. (1999) , and Ara u´jo (2002). In this study, the model by Carreira and Chu (1985) was modified to consider the fiber effects as 50 MPa 35 MPa 27 MPa 40 35 where fcf is the compressive strength of plain or AMFRC, eof is the strain corresponding to compressive strength of the materials used; rc and ec are the stress and strain on the curve, respectively; and b is a material parameter representing the influence of fibers in the descending branch of the stress–strain curve. It is noticed that the smaller value of b produces a less steep descending branch in the stress–strain curve. For a given value of compressive strength, eof is determined using Eqs. (5) and (6), and thus the only variable is b. Based on multi-linear regression analysis, the b value was defined as functions of fiber volume fraction,Vf (%), and the compressive strength of plain concrete, fc (MPa) b ¼ ( pp44ffifffiffficffifficffiðð11::17814 20:1:345V5f ÞVf Þfor for0 VVff \00::33 ð8Þ Figure 19 compares the compressive stress–strain curves obtained in the test results and those predicted by the proposed model. Due to the variation in compressive strength and strain at peak stress of different specimens, the stress–strain relationships were compared in a normalized manner. Figures 20a and 20b illustrate typical stress–strain curves from experimental results and proposed analytical model in the cases of 0.6 % amorphous metallic fiber volume fraction and concrete compressive strength of 35 MPa, respectively. The figures showed that the proposed model properly predicted the test results throughout all the curves. Thus, the proposed model is expected to be used for structural analysis of AMFRC for different compressive strengths and fiber volume fractions. 5. Conclusions In this paper, the characteristics of amorphous metallic fiber-reinforced concrete in fresh and hardened stages were ð7Þ investigated. The following conclusions can be drawn from the experimental results: • • • • • • The addition of amorphous metallic fibers to concrete reduces the workability according to fiber volume fraction and increases the air contents of fresh concrete mixture in a range of 0.4–0.6 %. Generally, amorphous metallic fibers don’t affect the compressive strength of concrete, the difference in compressive strength between the amorphous metal fiber-reinforced concrete and plain concrete is only 0.17–2.93 %. However, strain corresponding to peak stress, ultimate strain and toughness in compression were found to be increased proportionally to the fiber dosage in the concrete mix. The elastic modulus shows a decreasing tendency with increasing fiber volume fraction. The existing of amorphous metallic fibers in concrete matrix reduces the expansion of concrete in the horizontal direction, which contributes to the stabilization of Poisson’s ratio in high stress level. Equations to predict the modulus of elasticity and the strain at peak stress of amorphous metal fiber-reinforced concrete were proposed as functions of fiber volume fraction and the concrete compressive strength, and were validated by the test results. The prediction by the equations showed a good correlation with the test data. A stress–strain curve is also proposed to represent the complete stress–strain relationship based on multi-linear regression analysis method. The proposed modeling approach was found to present a good correlation with the experimental results. However, the application of the proposed model is limited to the test condition of this study: the maximum aggregate size of 13 mm and fiber length of 30 mm. Thus, further investigation and validation are necessary for different test conditions. Acknowledgments This research was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2014R1A1A2053499). Open Access This article is distributed under the terms of the Creative C o m m o n s A t t r i b u t i o n 4 . 0 I n t e r n a t i o n a l L i c e n s e (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. ACI 318- 14 ( 2014 ). Building code requirements for structural concrete . A¨ıtcin , P. C. , & Mehta , P. K. ( 1990 ). 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Ngoc-Hieu Dinh, Kyoung-Kyu Choi, Hee-Seung Kim. Mechanical Properties and Modeling of Amorphous Metallic Fiber-Reinforced Concrete in Compression, International Journal of Concrete Structures and Materials, 2016, 221-236, DOI: 10.1007/s40069-016-0144-9