Tensile fracture properties of an Ultra High Performance Fiber Reinforced Concrete (UHPFRC) with steel fiber

Composite Structures 92 (2010) 61–71

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Tensile fracture properties of an Ultra High Performance Fiber Reinforced Concrete (UHPFRC) with steel fiber Su-Tae Kang a, Yun Lee b,*, Yon-Dong Park c, Jin-Keun Kim d a

Department of Structural Material, Korea Institute of Construction Technology, 2311, Daehwa-dong, Ilsanseo-gu, Goyang 411-702, South Korea Department of Architecture, College of Engineering, Ewha Womans University, 11-1, Daehyun-dong, Seodaemun-gu, Seoul 120-750, South Korea c Faculty of Architecture and Civil Engineering, Daegu Haany University, 290, Yugok-dong, Gyeongsan, Gyeongbuk 712-715, South Korea d Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea b

a r t i c l e

i n f o

Article history: Available online 4 July 2009 Keywords: Steel fiber Fiber bridging Ultra High Performance Concrete Fiber Reinforced Concrete Inverse analysis Cracking

a b s t r a c t This paper presents a study of the tensile fracture properties of Ultra High Performance Fiber Reinforced Concrete (UHPFRC) considering the effects of the fiber content. To investigate the impact of fiber content, notched 3-point bending tests were executed, where the fiber volume ratio was varied from 0% to 5%. From the bending tests, it was found that the flexural tensile strength of UHPFRC linearly increases with increasing fiber volume ratio and the rule of mixture can be applied to UHPFRC. Furthermore, an inverse analysis was performed to determine the tensile fracture model of UHPFRC and a tri-linear tensile softening model is suggested. The suggested model successfully represents the increase of the stress-constant bridging zone and the decrease of the stress-resisting zone with increasing fiber content. The proposed model for various fiber content levels is simple and versatile and can be readily applied to structural design or numerical analysis of UHPFRC. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Exhibiting remarkable compressive strength and durability, concrete is one of the most widely adopted materials for the construction of bridges, together with steel. However, its inherent poor tensile and flexural strengths make it prone to cracking, and a gradual increase of brittleness according to an increase of compressive strength also brings numerous problems in its application. Recent researches on Ultra High Performance Concrete (UHPC) revealed that its mechanical characteristics include high tensile strength and large ductility continues to develop even after cracking in cooperation with fibers [1–4]. From these studies, Ultra High Performance Fiber Reinforced Concrete (UHPFRC), integrating UHPC and Fiber Reinforced Concrete (FRC), has emerged. Habel et al. [1], Richard and Cheyrezy [2] and Charron et al. [3] found that the elimination of coarse aggregates combined to the optimization of granular mixture in UHPFRC allows the acquisition of a homogenous and very dense cementitious matrix that offers ultra high strengths (compressive strength >150 MPa, tensile strength >10 MPa). With regard to tensile characteristics, Rossi et al. [4] found that the fiber of UHPFRC plays a critical role in the ductile behavior of a structure until

* Corresponding author. Tel.: +82 2 3277 6707; fax: +82 2 3277 2396. E-mail address: [email protected] (Y. Lee). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.06.012

flexural failure and reported the ultimate tensile strain capacity of UHPFRC up to 5  103. Recently, Wuest et al. [5] have developed a meso-mechanical model to predict the UHPFRC tensile response as a function of the volume, aspect-ratio, distribution and orientation of the fibers and the mechanical properties of the matrix. Fantilli et al. [6] proposed the multi-cracking approach for UHPFRC to achieve the multiple cracking and strain hardening phenomenon in uniaxial tension. In order to promote the applicability of UHPFRC structures, however, it is necessary to first develop design and structural analysis technologies in relation to the mechanical characteristics of UHPFRC. However, despite the recent efforts to find out the mechanical behaviors of UHPFRC, a lack of experimental data for quantitative assessment including information on the load sharing capacity of the fibers together with an absence of relevant methods to analyze the final state of UHPFRC structural members are currently impeding exploitation of the promising large deformation performance provided by this material. For these reasons, it is necessary to determine the tensile load carrying capacity and develop a reliable tensile fracture model through consideration of the effects of fibers on UHPFRC. Accordingly, this study examines experimentally the tensile fracture properties of UHPFRC and establishes a tension fracture model for accurate estimation of the deformability through a finite element analysis based on corresponding experimental data and results.

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Table 1 Mix design of UHPFRC. Cement content (kg/m3)

Relative weight ratios to cement Cement

Water

Silica fume

Fine aggregates

Filler

Superplasticizer

821.7 813.1 804.9 796.7 788.5 780.3

1.00

0.25

0.25

1.10

0.30

0.018

a

Steel fibera (Vf, %)

0 1 2 3 4 5

Fiber volume expressed as volumetric ratio to the whole volume.

Table 2 Physical and chemical properties of cement and silica fume. Material

Specific surface (cm2/g)

Density (g/cm3)

OPC Silica fume

3413 200,000

3.15 2.10

Chemical composition (%) SiO2

Al2O3

Fe2O3

CaO

MgO

SO3

21.01 96.00

6.40 0.25

3.12 0.12

61.33 0.38

3.02 0.1

2.3 –

Fig. 1. Specimen configuration and experimental setup.

Fig. 2. Failure configuration after test.

63

20

20

16

16

12

12

Load (kN)

Load (kN)

S.-T. Kang et al. / Composite Structures 92 (2010) 61–71

8

8

4

4

0

0 0

0.01

0.02

0.03

0.04

0.05

0

4

8

Displacement (mm)

16

20

(b) Vf : 1%

60

60

45

45

Load (kN)

Load (kN)

(a) Vf : 0%

30

30

15

15

0

0 0

4

8

12

16

0

20

4

8

12

16

20

Displacement (mm)

Displacement (mm)

(c) Vf : 2%

(d) Vf : 3%

80

80

60

60

Load (kN)

Load (kN)

12

Displacement (mm)

40

40

20

20

0

0 0

4

8

12

16

20

Displacement (mm)

(e) Vf : 4%

0

4

8

12

16

20

Displacement (mm)

(f) Vf : 5%

Fig. 3. Load–displacement experimental results.

2. Experimental program 2.1. Material The mix design of UHPFRC differs significantly from that of normal and high-strength concretes. UHPFRC mix compositions are characterized by high cement, superplasticizer, and silica fume content. The concrete mixture proportions applied in this study are tabulated in Table 1. The water–binder ratio is deter-

mined as w/b = 0.20, and a high percentage of silica fume (25% of cement weight) is implemented. Furthermore, in order to achieve sufficient strain-hardening behavior, various percentages of steel fibers in excess of 1% were incorporated. In this study, fiber content was selected as the main test variable and was classified into five groups corresponding to volume ratio, which was increased in increments of 1% from 1% to 5%. Generally, the use of filler through the partial replacement of cement provides enhanced strength at early age and durability. Therefore,

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80 Vf : 1 % Vf : 2 % Vf : 3 % Vf : 4 % Vf : 5 %

Load (kN)

60

40

20

0 0

4

8

12

16

20

Displacement (mm) Fig. 4. Load–displacement curves with respect to fiber volume ratio.

80

Flexural strength (MPa)

Experimental result Experimental result (Average) Linear fit (R2=0.97)

60

40

compact tension (CT) test, corresponding to those widely applied for ordinary concrete. Among them, the bending test is the most widely adopted method, owing to its simplicity. While the uni-axial tensile test offers the advantage of directly determining the tension softening curve, difficulties are encountered in securing accuracy. The CT test presents a noticeable advantage in being practically free from the effects of the self-weight of the specimen owing to the large failure area produced in specimens with small volumes. However, with this method the tension softening curve is determined indirectly, as with the bending test, and is rarely applied due to the need of special equipment. This study performed a 3-point bending test for the determination of the tension softening properties and curves of UHPFRC. For each concrete mixture shown in Table 1, five test specimens with dimensions of 100 mm  100 mm  400 mm are manufactured. The specimens were cut with a notch at mid-length using a diamond cutter prior to the execution of the test and after completion of concrete curing. The notch was set with a constant width of 4 mm. A universal testing machine with a capacity of 2000 kN was used for the bending test. Load was applied through displacement control at a speed of 1/1500 of the specimen span length (300 mm) per minute. One LVDT with a capacity of 10 mm was installed at both sides to measure the deflection of the center of the specimen during the test. A clip gage was attached at the bottom of the specimen to measure the crack width at the notch. Fig. 1 illustrates the characteristics of the bending test specimens and equipment. 2.3. Experimental results

20

0 0

1

2

3

4

5

Fiber volume ratio (%) Fig. 5. Variation of flexural strength with respect to fiber volume ratio.

siliceous filler was used in the concrete mix design. More specific material characteristics used in the concrete mix are as (1) Cement and reactive powder The cement and reactive powder adopted in this study are ordinary Portland cement (OPC) and silica fume, the physical and chemical properties of which are listed in Table 2. (2) Aggregates Fine aggregates with a density of 2.62 g/cm3 and sand with a mean particle size below 0.5 mm were utilized. Coarse aggregates were not used. (3) Superplasticizer Polycarboxylate superplasticizer (density 1.01 g/cm3, dark brown) was used. (4) Filler Siliceous filler with a mean particle size of 26.6 lm was applied. (5) Steel fibers High strength steel fibers (density 7.8 g/cm2, length 13 mm, diameter 0.2 mm, tensile strength 2500 MPa) were selected to improve toughness with respect to the tensile and flexural behavior.

2.2. 3-point bending test The test methods employed to determine the tension softening property of UHPFRC are the bending test, uni-axial tensile test, and

Fig. 2 shows a typical failure configuration after the 3-point bending test. After failure, it is shown that one large crack exists, accompanying fibers, which play an important role in bridging two crack faces. Due to the bridging mechanism of fibers, UHPFRC can provide superior performance especially under tension as compared to UHPC without fibers. Fig. 3 shows the experimentally obtained load–displacement curves with respect to fiber volume ratios. As seen in Fig. 3, higher fiber volume ratio results in larger scatter among the load–displacement curves for the same fiber volume. The large scatter with a high volume ratio is attributed to the intrinsic scatter at high strength levels and the degree of dispersion of fibers. Fig. 4 presents a comparison of load–displacement curves for five fiber volume ratios. It is revealed that the initial stiffness does not undergo a significant change with an increase of fiber content, while the maximum load increases gradually together with a gradual change to brittle behavior in the softening section. Although the structural ductility increases with fiber content for FRC, the displacement at peak load does not have an obvious trend with the fiber volume ratios in the case of UHPFRC. However, it is seen that UHPFRC reflects the superior behaviors of both UHPC and FRC with respect to high strength and ductility, respectively. The flexural strength of UHPFRC was obtained by Eq. (1).

rb ¼

3Pmax L 2bðh  a0 Þ2

ð1Þ

where Pmax is the maximum load; b and h the beam thickness and height, respectively; a0 the notch depth and L is the span length. With the development of FRC, the tensile performance of FRC in comparison with that of ordinary concrete has been studied by many researchers. Shah and Rangan [7], Naaman [8], Swamy et al. [9] and Mai [10] suggested the rule of mixture to consider the fiber content in calculating the flexural strength for FRC, given below as Eq. (2).

rbf ¼ Arbf 0 ð1  V f Þ þ BV f ðlf =df Þ

ð2Þ

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Pδ

σ 1

2

i+1 i

3 i σ(ω)

σi+1 Ο

ωi+1

ω

(a) Assumption of softening diagram

P

(b) Model for analysis

COD 3

PA

2 1

Ο

3 2 1

CODi+1

i

i+1 i

PE

δi+1

δ

(c) Fitting of P-δ

Ο

1

2

3 δi+1

δ

(d) Determination of crack width

Fig. 6. Poly-linear approximation method [13].

where rbf and rbf0 is the flexural strength with fiber and without fiber, respectively; Vf the fiber volume ratio; lf and df the length and diameter of fiber, respectively; and A and B is the experimental coefficients. Eq. (2) shows the linear dependence of the flexural strength (rbf) of FRC on the fiber volume ratio (Vf) and fiber shape (lf/df). Rearranging each term in Eqs. (2) and (3) can be obtained to clarify this linearity.

for flexural strength represented by Eq. (2) can be applied to UHPFRC with satisfactory confidence under the same fiber geometry conditions.

rbf ¼ V f ½Bðlf =df Þ  Arbf 0  þ Arbf 0

It is generally considered that flexural strength is a very important tensile property with respect to determining the fracturing capacity. However, this property alone is not sufficient to identify the whole fracture mechanism accompanying cracking initiation followed by tensile softening behavior due to the fiber bridging mechanism. In contrast with UHPC without fiber, the fiber in UHPFRC plays an important role in producing prominent bridging stress between opened crack faces. The bridging stress between the largely opened crack surfaces is the main source of the very high fracture toughness and ductility of UHPFRC. Accordingly, it is imperative to establish a tensile fracture model (bridging model) of UHPFRC. For the determination of the tensile fracture model of UHPFRC, the inverse analysis method suggested by Uchida and Kurihara [11] has been applied in this study. This method exploits the load– displacement curves obtained from flexural tensile tests to perform an inverse analysis using a FE analysis and the poly-linear approximation method proposed by Kitsutaka [12,13] and finally derives the tensile cohesive stress – crack opening displacement (COD) relationship. The poly-linear approximation method is an established technique, the accuracy of which has been sufficiently verified for ordinary concrete. Its accuracy with regard to high-strength concrete has also been verified by Kitsutaka [12,13]. Fig. 6 illustrates the general concept of the poly-linear approximation method proposed by Kitsutaka [13]. The outstanding performance of this

ð3Þ

For a given fiber geometry (lf/df), the flexural strength (rbf) is solely linearly dependent on the fiber volume ratio (Vf). It should, however, be verified that Eq. (3) for FRC is directly applicable to UHPFRC, which has an ultra-high concrete matrix strength and a relatively lower contribution of fiber than conventional FRC. The experiments performed in this study cover a large range of fiber volume ratios, from 1% up to 5%, and therefore the fiber content contribution to flexural strength can be examined with this equation. Fig. 5 shows the relationship between the average flexural strength and the corresponding fiber volume ratio obtained from the experiment. The flexural strength of UHPFRC is linearly dependent on the fiber content with high reliability, i.e. the coefficient of determination (R2) is 0.97. Formularizing this relationship, Eq. (4) is obtained as follows:

rbf ¼ 11:1V f þ rbf 0

ð4Þ

where rbf0 = 8.88 (MPa) and Vf is the fiber volume ratio by percentage. For the same fiber geometry (lf/df = 13/0.2) used in this study, comparing Eq. (4) with Eq. (3), the experimental coefficients A and B can be deduced as 1.0 and 0.307, respectively. From the above derivation of Eq. (4), it is concluded that the rule of mixture

3. Suggestion of tensile fracture model of UHPFRC 3.1. Procedure of inverse analysis

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Load (kN)

Load-displacement curve

Displacement (mm)

P

Number of elements = 1677 Number of nodes on crack line = 98

Fig. 9. Modelled tri-linear softening curve.

Cohesive stress (MPa)

notch

Primitive softening curve

Crack width (mm)

3.2. Stress-crack opening relationship from inverse analysis

Fig. 7. Procedure of inverse analysis.

method lies in the absence of need to assume the tensile softening curve, which should be appropriately pre-defined as input data of the FE analysis program. 25

In this method, the softening curve can be constructed point by point from the step-by-step optimum fit of the load–displacement curve, as depicted in Fig. 6. For UHPFRC, where the post-crack model is not fully established due to the high strength of the matrix and complicated bridging mechanism of fibers, the poly-linear approximation method can be very useful to predict the overall shape and behavior of the post-crack model. Fig. 7 depicts the overall procedure to derive the tensile cohesive stress-crack width relationship through an inverse FE analysis based on the load–displacement results of flexural tests. The finite element model is composed of 1677 triangular plane stress elements and 98 nodes on the crack line. The notch at the center of the specimens has been modeled in detail with dense meshing in order to precisely evaluate the crack propagation. As a result of the poly-linear approximation method, what is referred to as a primitive softening curve, which typically shows an abrupt softening – partial plateau and hardening – partial plateau and gradual softening behavior, is finally obtained. In the following section the primitive softening curve is simplified to model a softening curve having three linear regions.

As noted in the previous section, in our approach, the polylinear approximation method outputs a piecewise softening curve, called the primitive softening curve, for UHPFRC, through

Primitive softening curve Modelled softening curve

Cohesive stress (MPa)

20

15

10

5

0 0

2

4 Crack width (mm)

6

8 0

0.2

0.4

0.6

Crack width (mm)

Fig. 8. Modelling process of primitive softening curve by tri-linear regression.

0.8

1

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an internal step-by-step inverse algorithm. From the primitive softening curve, it is possible to predict the overall prototype of the tensile softening curve of UHPFRC. This prototype generally shows complex fracture behavior accompanying high strength matrix cracking, fiber bridging, and the relevant softening behavior. Fig. 8 shows the primitive softening curve obtained from the inverse analysis performed for UHPFRC specimens with five different fiber content levels. Typical primitive softening curves show an abrupt initial decrease of cohesive stress followed by the first flat branch with constant stresses. After some range of flatness, the cohesive stress increases slightly and maintains the second branch of constant stresses. Finally, the cohesive stress gradually decreases as the crack opens widely. Converting the aforementioned phenomena described by the primitive softening curve into mechanical terms, the stress-crack opening displacement relationship involves a series of initial softening-first plastic-instantaneous hardening-second plasticgradual softening behaviors. The first steep softening branch describes the bridging of the early microcracks and the formation of a macrocrack in the concrete matrix, and the two plastic branches accompanying slight hardening simulate the stress transfer relevant to the activation of the bridging mechanism be-

ft  f1 w þ ft for w < w1 w1 rcohesiv e ¼ f1 for w1 < w < w2 f1 rcohesiv e ¼  ðw  wc Þ for w2 < w < wc wc  w2 rcohesiv e ¼ 0 for wc < w

rcohesiv e ¼ 

80

Primitive analysis (R2=0.98) Modelled analysis (R2=0.98)

Calculated peak load (kN)

Calculated peak load (kN)

80

tween the matrix and fibers. As the crack width increases, final softening takes place due to the decrease of both the matrix cohesive stress and the bridging stress. Although the primitive softening curve itself is the optimum solution of an inverse analysis, there are serious problems in directly implementing that solution to a FE analysis or structural design, because the primitive softening curve is obtained as a type of discrete solution. Therefore, it is necessary to characterize the stress-crack opening relationship into a simplified form, which is feasible for structural design and nonlinear analysis of structural behaviors such as crack propagation and fracture. In this study, the primitive softening curve is modeled as a tri-linear softening curve that has an initial softening branch due to matrix cracking, and a bridging plateau region followed by a final softening branch, as depicted in Figs. 8 and 9 and given by Eq. (5).

60

40

20

Primitive analysis (R2=0.98) Modelled analysis (R2=0.98)

60

40

20

0

0 0

20 40 60 Measured peak load (kN)

0

80

(a) Comparison between case studies

20 40 60 Measured peak load (kN)

80

(b) Comparison between average values

Fig. 10. Comparison between measured and calculated peak load.

1.6 Primitive analysis (R2=0.71) Modelled analysis (R2=0.37)

Calculated displacement at peak (mm)

Calculated displacement at peak (mm)

1.6

1.2

0.8

Primitive analysis (R2=0.92) Modelled analysis (R2=0.98)

1.2

0.8

0.4

0.4 0.4

0.8

1.2

1.6

Measured displacement at peak (mm)

(a) Comparison between case studies

0.4

0.8

1.2

1.6

Measured displacement at peak (mm)

(b) Comparison between average values

Fig. 11. Comparison between measured and calculated displacement at peak.

ð5Þ

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where rcohesive is the cohesive stress (MPa) and w is the crack width (mm). In the modeling process of the primitive softening curve, two consecutive plateaus involving instantaneous hardening behavior are simplified into a single plateau having an appropriate value between the two plateaus by least square method. Through this simplification the modeled softening curve becomes very useful in numerical analyses or structural design while retaining physical

20

mechanisms such as plastic behavior due to fiber bridging. For verification of the obtained fracture model, peak loads obtained from the optimized softening curves, i.e. the primitive and modeled softening curves, for all the specimens are compared to the measured peak loads in Fig. 10, and excellent agreement between the measured and calculated results in the beam tests can be seen. The calculated and measured displacements at peak are compared in Fig. 11a, and the ratios of the measured-to-calculated displacement 40

Experiment Primitive analysis Modelled analysis

30

Load (kN)

Load (kN)

16

Experiment Primitive analysis Modelled analysis

12

8

20

10 4

0

0 0

4

8

12

0

16

4

(a) Vf : 1% 60

8

12

16

Displacement (mm)

Displacement (mm)

(b) Vf : 2% 60

Experiment Primitive analysis Modelled analysis

Experiment Primitive analysis Modelled analysis

40

Load (kN)

Load (kN)

40

20

20

0

0 4

8

12

0

16

4

8

Displacement (mm)

Displacement (mm)

(c) Vf : 3%

(d) Vf : 4%

80

Experiment Primitive analysis Modelled analysis

60

Load (kN)

0

40

20

0 0

4

8

12

16

Displacement (mm)

(e) Vf : 5% Fig. 12. Comparisons of load–displacement curves.

12

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at peak for modeled analysis have an average value of 0.96 and a corresponding standard deviation of 0.133. Taking the average value of five specimens in Fig. 11a, the ratios of the measured-to-calculated displacement at peak in Fig. 11b have a much smaller standard deviation of 0.045 with the same average value of 0.96. Additionally, it is seen in Fig. 12 that the load–displacement curves from simulations using the averaged primitive and modeled softening curves, which are optimized solutions, are comparable to those measured from the bending tests. In Fig. 12, it is seen that the primitive softening curve excellently simulates the measured load–displacement curve. Although it is natural that the primitive softening curve more accurately fits the measured results than the modeled softening curve, taking the simplicity of the latter into consideration, the overall agreement is quite good for the modeled softening curve despite some minor differences. 3.3. Effect of fiber volume ratio on tensile fracture model In the previous section, it was verified that the modeled softening curve with five parameters successfully simulates the overall tensile fracture behavior for various levels of fiber content. In this section, the effects of fiber volume ratios on the five parameters constituting Eq. (5) and Fig. 9 are discussed and relevant equations are suggested. Fig. 13 shows the relation between flexural strength and the strength parameters ft and f1. The linear relationships among the relevant average values shown in Fig. 13 can be expressed as Eq. (6), a simple linear equation.

ft ¼ arbf f1 ¼ brbf

ð6Þ

where rbf is the flexural strength obtained from the 3-point bending test and the experimental coefficients are obtained as a = 0.86 and b = 0.35. Although it cannot be verified that the strength parameter ft in Eq. (6) equates with the direct tensile strength of UHPFRC without performing a direct tensile test, the strength parameters ft and f1, which indicate the initiation of cracking and activation of bridging, can be approximately predicted as 86% and 35% of the flexural strength, respectively. Fig. 14 shows the variation of five softening curve parameters with respect to the fiber volume ratio. As can be expected by substituting rbf in Eq. (6) with Eq. (4), the strength parameters ft and f1 are linearly dependent on the fiber volume ratio, as shown in Fig. 14a and b. It is concluded that up to a 5% fiber volume ratio

Calculated strength (MPa)

80

ft f t (Average) f1 f1 (Average)

60

R 2=0.89

40 R2=0.99

20

0 0

20

40

60

80

Flexural strength (MPa) Fig. 13. Relationship between flexural strength and strength parameters ft and f1.

69

of UHPFRC, the strength parameters ft and f1 constituting the tensile fracture model delineated by Eq. (5) can be estimated from the fiber volume ratio with the given simple linear equation. Fig. 14c shows the variation of w1 with the fiber volume ratio. It appears that there is no significant trend with increasing fiber volume ratio up to 5%, for which w1 remains constant. Keeping w1 constant under variation of fiber volume is consistent with the fact that the initial branch determined by w1 (also ft and f1) of the tensile fracture model characterizes the bridging of the concrete matrix between the early microcracks, and the bridging is mainly due to the concrete matrix without fiber [14]. Meanwhile, the parameter w2 of the tensile fracture model shown in Fig. 9 determines the size of the bridging plateau region, where the fiber bridging mechanism between crack faces occurs while keeping cohesive stress constant. According to Fig. 14d, which shows the variation of w2  w1 with fiber volume ratio, the value of w2  w1, indicating the size of the stress-constant bridging zone, increases with a higher fiber volume ratio. In this study, the increasing trend up to 5% of fiber volume is approximated by an exponential function, which can describe the prominent reduction of the rate of increase above 3% fiber volume. The last parameter of the fracture model, wc, is the threshold value over which the cohesive stress can no longer be resisted by the fibers between crack faces. Fig. 14e shows the degradation of wc with increasing fiber volume ratio. The value of wc is related to the ductility of UHPFRC, and therefore Fig. 14e indicates that a high volume ratio of fiber does not always guarantee good ductility and there exists a threshold, which can be variant with other conditions (such as fiber length, shape, quality, concrete mix, placing method, workability, etc.). In formularizing the results in Fig. 14, in order to improve the reliability of the relationship between each parameter and the fiber volume ratio, an averaged value of five numerical results for each parameter is employed to derive the equations that consider the effect of the fiber volume ratio on each parameter. In this study, Eq. (7) is finally suggested as the tensile fracture model of UHPFRC taking into consideration the effect of fiber volume ratio.

ft ¼ aft V f þ bft f1 ¼ af1 V f þ bf1 w1 ¼ bw1 w2  w1 ¼ aw2 ð1  ebw2 V f Þ * l wc ¼ 2f for V f < bwc

ð7Þ

l

wc ¼ 2f eðV f bwc Þ þ awc ð1  eðV f bwc Þ Þ for V f P bwc where Vf is the fiber volume ratio by percent; lf the fiber length (13 mm in this study); experimental coefficients aft = 7.09, bft = 16.2, af1 = 3.79, bf1 = 3.69, bw1 = 0.0242, aw2 = 0.50, bw2 = 0.54, awc = 4.64, and bwc = 1.29 are obtained, respectively. Regarding the strength parameters ft and f1, averaged values of ft and f1 were fitted with linear functions of the volume ratio Vf. For w1, one experimental coefficient bw1 was obtained from a constant function fit of the averaged values. Differently from other parameters, it is seen that bw1 depends only on the concrete property, not on the fiber volume ratio. On the other hand, in Eq. (7), w2  w1 was fitted with an exponential growth function with the experimental coefficients aw2 and bw2. The size of the stress-constant bridging zone, w2  w1, was calibrated as zero when the fiber volume ratio is 0%. In this equation, aw2 is introduced to indicate the maximum threshold at a large Vf, and bw2 represents the growth rate of the bridging zone size with increasing Vf. Regarding wc related to ductility, the numerical results in Fig. 14e were fitted with constant and exponential decay functions, as shown in the last two equations of Eq. (7). Considering the

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bridging and pull-out mechanism of only each fiber rather than the bridging effect by concrete itself, the parameter wc indicating the size of the stress-resisting zone is severely affected by the fiber length lf. Therefore, in this study, it is considered that wc has a upper limit determined by the fiber half length lf/2. The degradation of wc is fitted by constant and exponential decay functions, as shown in Fig. 14e. The second term of the last equation in Eq.

25

Inverse analysis result Inverse analysis result (Average) Linear fit (R2=0.89)

50

20

40

15

f1 (MPa)

ft (MPa)

60

(7) represents the increasing deviation from the upper limit with an increasing volume ratio. The loss of ductility due to high fiber content can be explained by several factors such as difficulty in placing, the fiber balling effect, disturbance of the fiber distribution, and poor arrangement of fiber direction. Depending on these factors for a specific UHPFRC, the decrease of wc can be represented by adjustment of the experimental coefficients awc and bwc.

30

20

Inverse analysis result Inverse analysis result (Average) Linear fit (R2=0.95)

10

5

10

0

0

1 2 3 4 Fiber volume ratio (%)

5

0

1 2 3 4 Fiber volume ratio (%)

(a) Variation of ft

(b)Variation of f1

0.035

0.8 Inverse analysis result Inverse analysis result (Average) Exponential fit (R2=0.87)

0.03 0.6

w2 - w1 (mm)

0.025 0.02 0.015 0.01 0.005

0.4

0.2

Inverse analysis result Inverse analysis result (Average) Constant fit (w1 = 0.0242, σ,stdev=0.001)

0

0

0

1 2 3 4 Fiber volume ratio (%)

5

0

1 2 3 4 Fiber volume ratio (%)

(d) Variation of w2 − w1

(c) Variation of w1 8 7

1/

2

fiber length limit

6

wc (mm)

w1 (mm)

5

5 4 3 2

Inverse analysis result Inverse analysis result (Average) Constant-exponential fit (R2=0.88)

1 0

0

1 2 3 4 Fiber volume ratio (%)

5

(e) Variation of wc Fig. 14. Variation of fracture model parameters with respect to fiber volume ratio.

5

S.-T. Kang et al. / Composite Structures 92 (2010) 61–71

71

4. Conclusions

References

This study investigated the effects of fiber content on the tensile fracture behavior of UHPFRC and a numerical model is suggested based on the experimental results.

[1] Habel K, Viviani M, Denarié E, Brühwiler E. Development of the mechanical properties of an Ultra-High Performance Fiber Reinforced Concrete (UHPFRC). Cem Concr Res 2006;36(7):1362–70. [2] Richard P, Cheyrezy M. Composition of reactive powder concretes. Cem Concr Res 1995;25(7):1501–11. [3] Charron JP, Denarié E, Brühwiler E. Permeability of UHPFRC under high stresses. In: Proceedings of RILEM symposium, advances in concrete through science and engineering, Evanston, Il; March 2004. p. 12. [4] Rossi P, Arca A, Parant E, Fakhri P. Bending and compressive behaviors of a new cement composite. Cem Concr Res 2005;35(1):27–33. [5] Wuest J, Denarié E, Brühwiler E. Model for predicting the UHPFRC tensile hardening response. In: Proceedings of 2nd international symposium on ultra high performance concrete, University of Kassel, Germany; 2008. p. 153–60. [6] Fantilli AP, Mihashi H, Vallini P. Multi-cracking phenomenon of HPFRCC in tension. In: Proceedings of 8th international symposium on utilization of high-strength and high-performance concrete, Tokyo, Japan; 2008. p. 555– 62. [7] Shah SP, Rangan VB. Fiber reinforced concrete properties. J ACI 1971;68:126–35. [8] Naaman AE. A statistical theory of strength for fiber reinforced concrete. PhD Thesis, Massacusetts Institute of Technology; 1972. [9] Swamy RN, Mangat PS, Rao CVSK. The mechanics of fiber reinforcement of cement matrices. In: An international symposium: fiber reinforced concrete; 1974. p. 1–28. [10] Mai YM. Strength and fracture properties of asbestos-cement mortar composites. J Mater Sci 1979;14:2091–102. [11] Uchida Y, Kurihara N. Determination of tension softening diagrams of various kinds of concrete by means of numerical analysis. In: FRAMCOS-2, Germany; 1995. [12] Kitsutaka Y. Fracture parameters for concrete based on poly-linear approximation analysis of tension softening diagram. In: FRAMCOS-2, Germany; 1995. [13] Kitsutaka Y. Fracture parameters by poly-linear tension softening analysis. J Eng Mech 1997;123(5):444–50. [14] Wittmann FH, Rokugo K, Brühwiler E, Mihashi H, Simonin P. Fracture energy and strain softening of concrete as determined by means of compact tension specimens. Mater Struct 1988;21:21–32.

1. From notched 3-point bending tests, it is observed that the flexural tensile strength of UHPFRC linearly increases as the fiber volume ratio increases from 0% to 5% and the rule of mixture holds for the flexural strength of UHPFRC. 2. The inverse analysis method for determination of a tensile fracture model of UHPFRC is adopted. From a preliminary analysis, a primitive softening curve is obtained and a tri-linear softening curve is suggested based on the primitive curve. 3. Five parameters constituting the suggested model are obtained and each parameter is characterized up to a 5% fiber volume ratio. From the parametric study, it is found that the strength parameters are linearly dependent on fiber content and the initial decreasing branch size of the suggested model remains constant for various levels of fiber content. Furthermore, the suggested model is devised to successfully represent the increase of the stress-constant bridging zone and the decrease of the stress-resisting zone with increasing fiber content. Acknowledgments This study has been a part of a research project supported by Korea Ministry of Education, Science and Technology (MEST) via the research group for control of crack in concrete. The authors wish to express their gratitude for the financial support that made this study possible.