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Cement and Concrete Composites journal homepage: www.elsevier.com/locate/cemconcomp

Fiber volume fraction and ductility index of concrete beams Alessandro P. Fantilli*, Bernardino Chiaia, Andrea Gorino Department of Structural, Building and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 May 2015 Received in revised form 2 October 2015 Accepted 9 October 2015 Available online 19 October 2015

The mechanical response of ﬁber-reinforced concrete (FRC) beams depends on the amount of ﬁbers, and the transition from brittle to ductile behavior in bending is related to a critical value of ﬁber volume fraction. Such quantity, which is mechanically equivalent to the minimum amount of steel rebars in reinforced concrete beams, can be deﬁned according to the new approach proposed herein. It derives from the application of a general model and from the introduction of the so-called ductility index (DI). When FRC beams show a ductile behavior DI is positive, whereas DI is negative in the case of brittle response. Both the theoretical and experimental results prove the existence of a general linear relationship between DI and the ﬁber volume fraction. Accordingly, a new design-by-testing procedure can be used to determine the critical value of ﬁber volume fraction, which corresponds to a ductility index equal to zero. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Fiber-reinforced concrete Beams Bending moment Deﬂection-hardening Ultimate limit state Ductility index

1. Introduction Depending on the ﬁber volume fraction Vf used to reinforce the cementitious matrix, ﬁber-reinforced concrete (FRC) ties behave differently (Fantilli et al. [1], Naaman [2]). Speciﬁcally, in the case of low values of Vf, the tensile force F after cracking (which occurs at the cracking load Fcr), remains lower than Fcr as the elongation DL increases (see Fig. 1a). Such ties fail in a brittle manner, and the socalled strain-softening occurs in the presence of a single crack. Conversely, with an higher amount of ﬁbers, and after the softening stage subsequent to the growth of the ﬁrst crack, F increases and reaches the ultimate load Fu > Fcr. This is the case of the ductile response, in which the ties show strain-hardening behavior and multiple cracking. At the transition from the brittle to the ductile response, when Fu ¼ Fcr (Fig. 1a), the corresponding critical value of Vf can be considered as the minimum amount of ﬁbers that guarantees the strain-hardening behavior of the FRC ties (Fantilli et al. [1]). Similarly, the behavior of FRC beams in three point bending, as illustrated in Fig. 1b in terms of applied load P vs. midspan deﬂection d, is also a function of Vf (Naaman [2]). At the ﬁrst relative maximum (when P ¼ Pcr*), the effective cracking takes place,

* Corresponding author. E-mail addresses: [email protected] (A.P. Fantilli), [email protected] polito.it (B. Chiaia), [email protected] (A. Gorino). http://dx.doi.org/10.1016/j.cemconcomp.2015.10.019 0958-9465/© 2015 Elsevier Ltd. All rights reserved.

whereas at ultimate load Pu (the second relative maximum) strain localization occurs in the tensile zone. The value of Pcr* is in turn higher than the ﬁrst cracking load (i.e., Pcr), which corresponds to the attainment of the tensile strength in the bottom edge of the FRC beam (Fig. 1b). For low amounts of ﬁbers, Pu is always lower than Pcr*, and the brittle response of the beams (called deﬂectionsoftening) is evidenced by the presence of a single crack. Conversely, both the deﬂection-hardening (i.e., Pu > Pcr* in Fig. 1b), and the presence of more than one crack as well, are the results of tests performed on FRC beams containing high amounts of ﬁbers. Consequently, at the transition from brittle to ductile behavior, when Pu ¼ Pcr* in Fig. 1b, the minimum ductility is attained and a critical amount of ﬁbers can be deﬁned. As this quantity of ﬁbers has the same mechanical role of the minimum area of steel reinforcing bars in reinforced concrete beams (Fantilli et al. [3]), it can be deﬁned as the minimum amount of ﬁber-reinforcement (i.e., Vf ¼ Vf,min in Fig. 1b). A univocal and simple approach able to predict Vf,min in FRC beams does not exist, despite the huge number of tests available in the current literature. For instance, Naaman [2] proposed a formula to compute Vf,min based on the equation Pcr* ¼ Pu, where the values of both the loads are functions of the ﬂexural and of the residual tensile strengths of FRC, respectively. However, the relationship between the strengths and the ﬁber volume fraction cannot be easily and univocally deﬁned, hence the analytical prediction of Vf,min is not always effective. On the other hand, the approaches suggested by code rules are even more complicated, due to the

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Fig. 1. The behavior of FRC structures as a function of the ﬁber content: (a) axial load vs. elongation diagram of a tie; (b) applied load vs. midspan deﬂection of a beam in three point bending.

large experimental campaigns required for the deﬁnition of Vf,min. In particular, Model Code 2010 (ﬁb [4]) ﬁrstly recommends the classiﬁcation of FRC, and the evaluation of the residual strengths, by means of three point bending tests on notched beams. Then, the displacements measured in a second series of tests, performed on full-scale FRC elements in bending, with different contents of ﬁbers, are needed. The ductility requirement in bending (and the corresponding Vf Vf,min) is satisﬁed when the ultimate or the peak displacements are sufﬁciently large (Caratelli et al. [5], de la Fuente et al. [6]). With the aim of simplifying the evaluation of Vf,min, a new design-by-testing procedure, capable of predicting the brittle/ ductile behavior of FRC beams in bending, is proposed in the following. It is the result of both the theoretical and experimental investigations described in the next sections. 2. General model A multi-scale general model is introduced herein to predict the behavior of the FRC beam depicted in Fig. 1b. The ﬁberreinforcement is modeled with an ideal tie (Fig. 2a), composed by a straight ﬁber and the surrounding cementitious matrix, having a single orthogonal crack in the midsection. The pullout mechanism of this element provides the stress-strain relationship of the cracked FRC. Only when this relationship is known, can the mechanical response of the FRC beams in bending be properly deﬁned. 2.1. Modeling the ﬁber pullout The ideal tie illustrated in Fig. 2a has a square cross-section, in

which the area of the cementitious matrix Ac is a function of the amount of ﬁbers used in the FRC beam:

Ac ¼

Af p$f2 ¼ Vf 4$Vf

(1)

where Af, f ¼ area and diameter of the ﬁber cross-section, respectively. The portion of the tie delimited by the cracked cross-section (in the midspan) and the so-called Stage I cross-section (where the perfect bond between steel and concrete is re-established) is investigated. Within this block of length ltr (¼ transfer length), as the horizontal coordinate z increases, stresses move from steel to concrete in tension, due to the bond-slip mechanism acting at the interface of the materials. Such slip s vanishes in the Stage I crosssection (Fig. 2b), where stresses (of ﬁber sf,I and of concrete sc,I in Fig. 2c) are computed with the well-known linear elastic formulae, under the hypothesis of perfect bond between the materials:

sf;I ¼ n$

sc;I ¼

N Ac þ n$Af

N Ac þ n$Af

(2)

(3)

where n ¼ Ef/Ec ¼ ratio between the Young's moduli of the ﬁber and of the cementitious matrix; N ¼ axial load applied to the ideal tie (Fig. 2a). Within the transfer length ltr, the interaction between ﬁber and matrix is described by the following equilibrium and compatibility

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141

In Eq. (7), the value of the slip s0 in the cracked cross-section is equal to the half of crack width w, whereas Eq. (8) states the absence of slips at a distance ltr from the mid-section. Only when ltr < Lf/2 (Lf ¼ length of the ﬁber), can this condition be considered valid. Finally, Eq. (9) and Eq. (10) impose the Stage I conditions [Eq. (2) and Eq. (3)] for the state of stress in ﬁber and matrix, respectively. According to Model Code 2010 (ﬁb [4]), the mean tensile strength of concrete fct can be estimated from the compressive strength fc (expressed in MPa):

fct ¼ 0:3$ðfc 8 Þ2=3

for fc 58 MPa

fct ¼ 2:12$lnð 1 þ 0:1$fc Þ

for fc > 58 MPa

(11.a) (11.b)

The residual tensile stress on the crack surfaces of the ideal tie (sc0 in Fig. 2c) can be deﬁned by the “ﬁctitious crack model” shown in Fig. 3a. It consists of a bilinear stress-crack opening displacement relationship, sc e w, as proposed by Model Code 2010 (ﬁb [4]):

w sc ðwÞ ¼ fct $ 1:0 0:8$ w1

for 0 < w w1

w sc ðwÞ ¼ fct $ 0:25 0:05$ w1

Fig. 2. Modeling the ﬁber pullout: (a) the ideal tie composed by a straight ﬁber and the surrounding cementitious matrix in presence of a single crack; (b) slip between ﬁber and matrix; (c) stresses in the ﬁber and in the matrix.

dsf p 4 ¼ f $t½sðzÞ ¼ $t½sðzÞ f dz Af

(4)

" # ds sf ðzÞ sc ðzÞ ¼ dz Ef Ec

(5)

where sf, sc ¼ stress in ﬁber and matrix, respectively (Fig. 2c); pf ¼ perimeter of the ﬁber cross-section; and t ¼ bond stress corresponding to the slip s between the materials (Fig. 2b). The resultant of axial stresses, acting in each cross-section of the tie, can be computed as:

N ¼ sf $Af þ sc $Ac

(12.b)

where w1 ¼ GF/fct; wc ¼ 5$GF/fct; and GF ¼ 0:073$fc0:18 ¼ fracture energy of concrete in tension (fc in MPa). In the same way, the interaction between ﬁber and matrix (Fig. 2b) needs the deﬁnition of a bond-slip t e s relationship. For the sake of simplicity, the model proposed by Fantilli and Vallini [8], originally developed for smooth steel ﬁbers in a cementitious matrix, is adopted (see Fig. 3b):

t ¼ tmax $

a s s1

for 0 s < s1

t ¼ tf þ tmax tf $eb$ðs1 sÞ

equations, according to Chiaia et al. [7]:

for w1 < w wc

(12.a)

(13.a)

for s1 s

(13.b)

where tmax ¼ maximum bond stress; tf ¼ residual bond stress; s1 ¼ 0.1 mm; a ¼ 0.5; and b ¼ 2/mm. The value of tf, which is a function of the compressive strength (in MPa), and that of tmax, which also depends on the ﬁber diameter (in mm), can be evaluated with the following formulae (Fantilli and Vallini [8]):

pﬃﬃﬃﬃ tf ¼ 0:1$ fc

(14)

pﬃﬃﬃﬃ 1:572 tmax ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ$ fc 12:5 þ f

(15)

(6)

To solve the system composed by the Eqs. (4)e(6), the following boundary conditions are needed:

w s0 ¼ 2

(7)

sðz ¼ ltr Þ ¼ 0

(8)

sf ðz ¼ ltr Þ ¼ sf;I

(9)

sc ðz ¼ ltr Þ ¼ sc;I

(10)

2.2. Numerical evaluation of N (w) The pullout model previously described, and referred to the ideal tie of Fig. 2, consists of the so-called “tension-stiffening” problem, which has to be solved within the one-dimensional domain of length ltr (Fig. 2a). The proposed solution is based on the following iterative procedure (see the ﬂow-chart depicted in Fig. 4): 1. Assign a value to the crack width w in the midsection of the ideal tie (Fig. 2a).

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Fig. 3. Residual stress on the crack surface and bond stress at the interface of ﬁber and matrix: (a) ﬁctitious crack model proposed by Model Code 2010 (ﬁb [4]); (b) bond-slip model proposed by Fantilli and Vallini [8].

2. Assume a trial value to the axial load N (Fig. 2a). 3. Compute the slip s0 in the midsection (z ¼ 0 in Fig. 2b) with Eq. (7). 4. Calculate the tensile stress of the matrix sc0 in the midsection (z ¼ 0 in Fig. 2c) by means of Eq. (12). 5. According to Eq. (6), the tensile stress of the ﬁber in the midsection (z ¼ 0 in Fig. 2c) can be evaluated with the following equation:

sf0 ¼

N sc0 $Ac Af

(16)

In order to exclude the failure of the ﬁber, and the subsequent brittle response of the ideal tie, sf0 must be lower than fu (where fu ¼ tensile strength of the ﬁber). 6. In the Stage I cross-section (z ¼ ltr in Fig. 2c), compute sf,I with Eq. (2) and sc,I with Eq. (3). 7. Consider Dl as a small part of the unknown ltr < Lf/2, and deﬁne zi ¼ i ∙ Dl (where i ¼ 1, 2, 3, …). 8. For each i (or zi) calculate: - The bond stress ti, related to the slip si1 [Eq. (13)]; - The stress sf,i in the ﬁber, by using Eq. (4) written in the ﬁnite difference form:

4 sf;i ¼ sf ;i 1 $ti $Dl f

(17)

- The stress sc,i in the matrix according to Eq. (6):

sc;i ¼

N sf;i $Af Ac

(18)

according to Chiaia et al. [9], the crack opening displacement w can be smeared into the equivalent strain εF as follows (Fig. 5b):

εF ¼

sc ðwÞ w þ Ec Lf

In other words, Lf is assumed to be the characteristic length of the FRC in tension, where the inelastic strains localize. Similarly, the axial load N of Fig. 5a, applied to the cross-sectional area of the tie (i.e., Af þ Ac in Fig. 2a), turns into the tensile stresses sF of the cracked FRC (Fig. 5b):

sF ¼

N Ac þ Af

(21)

2.3. Modeling the beam response The behavior of the FRC beam illustrated in Fig. 1b can also be deﬁned by the moment M vs. curvature m relationship of a single cross-section. In fact, the cross-sectional M - m curves and the P d diagrams (of Fig. 1b) have similar shapes, both depending on the amount of ﬁbers Vf. As a consequence, the ductility requirement in bending can be assessed by comparing the effective cracking moment Mcr* and the ultimate bending moment Mu, which occur in the midspan cross-section of the FRC beam when P ¼ Pcr* and P ¼ Pu (see Fig. 1b), respectively. As depicted in Fig. 6, to evaluate the moment-curvature relationship of a beam cross-section, having a width B and a depth H (see Fig. 6b), the classical hypothesis of linear strain proﬁle is adopted (Fig. 6c):

εF ¼ l þ m$y - The slip si by means of the ﬁnite difference form of Eq. (5):

!

si ¼ si1

sf ;i sc;i $Dl Ef Ec

(19)

9. When si ¼ 0 [Eq. (8)], if sf,i s sf,I or sc,i s sc,I [Eq. (9) and Eq. (10), respectively], change N and go back to step 3. For a given w, such procedure calculates the corresponding axial load N. Therefore, the complete N - w curve illustrated in Fig. 5a can be obtained by varying the assigned crack width. Then, the mechanical response of the cracked FRC in tension (when the strain in FRC εF is higher than that at cracking εF1 ¼ fct/Ec in Fig. 5b) is deﬁned on the base of the pullout response previously computed. Indeed,

(20)

(22)

where εF ¼ strain in a generic zone of the cross-section; l ¼ parameter corresponding to the strain at the origin of the coordinate y (located at H/2 from the edges of the cross-section e Fig. 6b). In agreement with Chiaia et al. [9], in the absence of an external axial load, the resultant R of the cross-sectional stresses becomes: þH=2 Z

R ¼ B$

sF dy ¼ 0

(23)

H=2

where sF ¼ stress in a generic zone of the FRC cross-section. The internal bending moment M, corresponding to a given state of stress, can be computed as follows [9]:

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143

Fig. 5. The behavior of FRC in tension: (a) axial load vs. crack width diagram of an ideal tie; (b) stress-strain relationship of the ﬁber-reinforced composite.

needs to be deﬁned. It is composed by the ascending branch of the Sargin's parabola (ﬁb [4]) in compression, and by the linear elastic law in tension (Fig. 7):

sF ¼ fc $ sF ¼ Ec $εF

k$h h2 1 þ ðk 2Þ$h for 0 εF < εF1

for εc1 εF < 0

(25.a) (25.b)

where k ¼ Ec/Ec1 ¼ plasticity number (i.e., the ratio between the tangent modulus of elasticity at the origin of the stress-strain diagram Ec, and the secant modulus from the origin to the peak compressive stress Ec1); h ¼ j εF/εc1 j ¼ compressive strain normalized with respect to εc1 (¼ strain at the peak of stress in compression). The complete stress-strain sF e εF relationship of the FRC, including both the uncracked and the cracked stages, is reported in Fig. 7.

2.4. Numerical evaluation of the moment-curvature relationship Fig. 4. Flow-chart for the computation of the ﬁber pullout.

Due to the nonlinear response of the FRC (Fig. 7), the crosssectional M - m relationship of the beams in bending (Fig. 6a) can be deﬁned through the following iterative procedure (see the corresponding ﬂow-chart in Fig. 8):

þH=2 Z

M ¼ B$

sF $y dy

(24)

H=2

To obtain the state of stress in the FRC cross-section, also the mechanical response of the uncracked FRC (when - εc1 εF < εF1)

1. Assign a value to the curvature m (Fig. 6c). 2. Assume a trial value for the strain parameter l (Fig. 6c). 3. Divide H in m stripes of depth Dh ¼ H/m (Fig. 6b), and deﬁne the ordinate yj (where j ¼ 1, 2, … m):

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Fig. 6. Modeling FRC beams in bending: (a)e(b) geometry of the three point bending beam; (c) strain proﬁle in a cross-section; (d) diagram of bending moment; (e) crosssectional moment-curvature diagram.

Fig. 8. Flow-chart for the computation of the moment-curvature relationship. Fig. 7. Complete stress-strain relationship of FRC.

H yj ¼ þ j$Dh 2

3. Deﬁnition of the ductility index

(26)

4. For each j (or yj) calculate - The strain εF,j with Eq. (22); - The stress sF,j (corresponding to εF,j) by means of the nonlinear relationship depicted in Fig. 7. 5. Compute the resultant R of the cross-sectional stresses [Eq. (23)]. 6. If R s 0, then change l and go to step 4. 7. Calculate the internal bending moment M [Eq. (24)] For a given m, such procedure provides the corresponding bending moment M. Hence, the complete M - m diagram illustrated in Fig. 6e can be obtained by varying the assigned curvature.

The brittle/ductile behavior of 54 ideal FRC beams in bending (Fig. 1b) is herein assessed by means of the M - m relationship previously introduced. The FRC specimens, having B ¼ H/2 (Fig. 6), are divided into 18 groups of 3 beams, with the same geometrical and material properties, but different ﬁber volume fractions Vf. Two depths (H ¼ 200 and 400 mm) and three classes of cementitious matrix (fc ¼ 30, 45 and 60 MPa) are taken into account. The reinforcement of each group consists of steel ﬁbers (Lf ¼ 60 mm, fu ¼ 1000 MPa, and Ef ¼ 210,000 MPa) having one of the following aspect ratios: Lf/f ¼ 40, 60, and 80. A single ideal FRC beam is identiﬁed by the alphanumeric label SX_CYY_AZZ_W, where X is related to the beam depth (X ¼ 1 when H ¼ 200 mm, and X ¼ 2 when H ¼ 400 mm), YY is the compressive strength (in MPa), ZZ is the ﬁber aspect ratio, and W is a number (1, 2, or 3) associated to Vf. All the mechanical and geometrical properties of the beams are reported in Table 1 (for the beams S1) and Table 2 (for the beams S2).

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145

Table 1 Properties of the ideal beams S1. Group

Beam

H (mm)

fc (MPa)

Lf/f

Vf (%)

Mcr* (kNm)

Mu (kNm)

DI

Vf,min (%)

v

1

S1_C30_A80_1 S1_C30_A80_2 S1_C30_A80_3 S1_C30_A60_1 S1_C30_A60_2 S1_C30_A60_3 S1_C30_A40_1 S1_C30_A40_2 S1_C30_A40_3 S1_C45_A80_1 S1_C45_A80_2 S1_C45_A80_3 S1_C45_A60_1 S1_C45_A60_2 S1_C45_A60_3 S1_C45_A40_1 S1_C45_A40_2 S1_C45_A40_3 S1_C60_A80_1 S1_C60_A80_2 S1_C60_A80_3 S1_C60_A60_1 S1_C60_A60_2 S1_C60_A60_3 S1_C60_A40_1 S1_C60_A40_2 S1_C60_A40_3

200

30

80

30

60

200

30

40

200

45

80

200

45

60

200

45

40

200

60

80

200

60

60

200

60

40

3.06 3.29 3.56 3.10 3.28 3.49 3.21 3.44 3.69 3.87 4.09 4.31 3.91 4.10 4.29 4.05 4.26 4.47 4.54 4.71 4.88 4.58 4.72 4.87 4.69 4.87 5.05

2.39 3.61 4.86 2.43 3.33 4.27 2.76 3.69 4.65 2.74 4.30 5.85 2.77 3.93 5.11 3.18 4.36 5.55 3.08 4.89 6.69 3.10 4.47 5.83 3.57 4.94 6.32

0.22 0.10 0.36 0.22 0.02 0.22 0.14 0.07 0.26 0.29 0.05 0.36 0.29 0.04 0.19 0.21 0.02 0.24 0.32 0.04 0.37 0.32 0.05 0.20 0.24 0.02 0.25

0.44

200

0.30 0.50 0.70 0.40 0.60 0.80 0.70 1.00 1.30 0.30 0.50 0.70 0.40 0.60 0.80 0.70 1.00 1.30 0.30 0.50 0.70 0.40 0.60 0.80 0.70 1.00 1.30

0.68 1.13 1.58 0.67 1.01 1.35 0.77 1.11 1.44 0.63 1.05 1.47 0.63 0.94 1.25 0.72 1.02 1.33 0.62 1.03 1.45 0.62 0.93 1.24 0.71 1.01 1.32

2

3

4

5

6

7

8

9

0.59

0.90

0.48

0.64

0.98

0.48

0.65

0.99

Table 2 Properties of the ideal beams S2. Group

Beam

H (mm)

fc (MPa)

Lf/f

Vf (%)

Mcr* (kNm)

Mu (kNm)

DI

Vf,min (%)

v

10

S2_C30_A80_1 S2_C30_A80_2 S2_C30_A80_3 S2_C30_A60_1 S2_C30_A60_2 S2_C30_A60_3 S2_C30_A40_1 S2_C30_A40_2 S2_C30_A40_3 S2_C45_A80_1 S2_C45_A80_2 S2_C45_A80_3 S2_C45_A60_1 S2_C45_A60_2 S2_C45_A60_3 S2_C45_A40_1 S2_C45_A40_2 S2_C45_A40_3 S2_C60_A80_1 S2_C60_A80_2 S2_C60_A80_3 S2_C60_A60_1 S2_C60_A60_2 S2_C60_A60_3 S2_C60_A40_1 S2_C60_A40_2 S2_C60_A40_3

400

30

80

30

60

400

30

40

400

45

80

400

45

60

400

45

40

400

60

80

400

60

60

400

60

40

24.47 26.30 28.52 24.77 26.24 27.92 25.71 27.51 29.53 30.94 32.76 34.50 31.29 32.76 34.30 32.36 34.09 35.73 36.28 37.70 39.08 36.61 37.76 38.97 37.52 38.95 40.36

19.12 28.88 38.89 19.40 26.67 34.20 22.07 29.51 37.19 21.91 34.38 46.82 22.13 31.47 40.90 25.42 34.84 44.41 24.62 39.14 53.50 24.81 35.72 46.62 28.57 39.55 50.56

0.22 0.10 0.36 0.22 0.02 0.22 0.14 0.07 0.26 0.29 0.05 0.36 0.29 0.04 0.19 0.21 0.02 0.24 0.32 0.04 0.37 0.32 0.05 0.20 0.24 0.02 0.25

0.44

400

0.30 0.50 0.70 0.40 0.60 0.80 0.70 1.00 1.30 0.30 0.50 0.70 0.40 0.60 0.80 0.70 1.00 1.30 0.30 0.50 0.70 0.40 0.60 0.80 0.70 1.00 1.30

0.68 1.13 1.58 0.67 1.01 1.35 0.77 1.11 1.44 0.63 1.05 1.47 0.63 0.94 1.25 0.72 1.02 1.33 0.62 1.03 1.45 0.62 0.93 1.24 0.71 1.01 1.32

11

12

13

14

15

16

17

18

As an example, the M - m diagrams of the beams S1_C45_A60_1 and S1_C45_A60_3 are reported in Fig. 9a and Fig. 9b, respectively. Two stationary points, corresponding to the effective cracking moment (Mcr*) and the ultimate bending moment (Mu), are clearly evident in both the ﬁgures. Fig. 9a shows the so-called deﬂectionsoftening response (i.e., Mu < Mcr* and the FRC beam is underreinforced), whereas the M - m diagram illustrated in Fig. 9b represents the typical deﬂection-hardening behavior (i.e., Mu > Mcr* and the ﬁber-reinforcement is higher than Vf,min). The values of Mcr*

0.59

0.90

0.48

0.64

0.98

0.48

0.65

0.99

and Mu, taken on the M - m diagrams of the 54 ideal FRC beams, are collected in Tables 1 and 2. According to Fantilli et al. [10], the ductile behavior of FRC beams corresponds to a positive value of the following ductility index (DI):

DI ¼

Mu Mcr Pu Pcr ¼ Mcr Pcr

(27)

As DI < 0 in beams showing a deﬂection-softening, the minimum amount of ﬁbers Vf,min (or, equivalently, the minimum

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Fig. 9. Application of the general model to beams of group 5 (see Table 1): (a) M - m diagram of the beam S1_C45_A60_1; (b) M - m diagram of the beam S1_C45_A60_3; (c) DI - Vf relationship and deﬁnition of Vf,min.

required ductility) can be computed by imposing DI ¼ 0. Tables 1 and 2 report the values of DI calculated for the ideal beams investigated herein. For each group of beams (e.g., those of group 5 in Fig. 9c), a linear relationship between DI and Vf is

attained. Thus, the values of Vf,min, detected for each group with the intersection between the line DI - Vf and the horizontal axis (i.e., DI ¼ 0), are reported in Table 1 (for the beams S1) and Table 2 (for the beams S2).

Table 3 Properties of FRC beams tested in some experimental campaigns. Group

Beam

I

N2 N3 N4 F80/60_Cf10 F80/60_Cf20 F80/60_Cf30 Mix_1_1 Mix_1_2 Mix_1_3 Mix_2_1 Mix_2_2 Mix_2_3 Mix_3_1 Mix_3_2 Mix_3_3 50C(40) 50C(80) 75C(40) 75C(80) C25/30(25) C25/30(75) 0.20(A) 0.52(A) 0.65(A) 0.91(A) S13-HL-28d S26-HL-28d S39-HL-28d 4P-LN-20 4P-LN-40 DWP_0.2 DWP_0.5 DWP_1 DWP_1.5 FP_0.2 FP_0.5 FP_1 FP_1.5

II

III

IV V VI VII

VIII

IX X

XI

a b

H (mm)

fc (MPa)

Lf/f

Vf (%)

Fiber

References

100

100

50

Banthia and Gupta [15]

125a

H_S

Barros et al. [12]

1000

100

21.5

113

0.50 0.75 1.00 0.13 0.25 0.38 0.36

F_S

150

97.4 92.6 86.6 33.0b

S_P

Fantilli et al. [10]

H_S

Jones et al. [13]

B (mm)

80

23.4

0.74

22.9

1.10

100

50

54.0b

60

100

75

54.0b

60

b

65

50.2b 48.7b 51.7b 52.7b 28

60

a

150

125

150

150

150

150

150

150

100

100

42.7b 48.8b e

100

100

e

Depth of the ligament of a notched beam. Estimated mean cylindrical compressive strength.

33.0

167

80 150

89

0.51 1.02 0.51 1.02 0.32 0.96 0.20 0.52 0.65 0.91 0.17 0.33 0.50 0.25 0.51 0.20 0.50 1.00 1.50 0.20 0.50 1.00 1.50

H_S H_S

Lee and Barr [14]

U_S

Michels et al. [16]

H_S

Mobasher et al. [17]

H_S

Soetens and Matthys [18]

S_P

Wu [19]

S_P

A.P. Fantilli et al. / Cement and Concrete Composites 65 (2016) 139e149

If the normalized reinforcement ratio v ¼ Vf/Vf,min is introduced (Fig. 10a), the existence of a linear function DI - v can be argued and the results can be extended to all the FRC beams. This line passes through the point corresponding to Vf,min (i.e., v ¼ 1, DI ¼ 0) and has a well-deﬁned slope. By applying the least square approximation to all the couples of DI and v values, previously computed for the 54 ideal FRC beams, this slope is computed and results equal to 0.7 (see Fig. 10a). In other words, for all the groups of FRC beams, the evaluation of the ductility index can be performed by means of the following general equation:

DI ¼ 0:7$ðv 1Þ

(28)

4. Experimental results compared with the predictions of the ductility index To verify the accuracy of the proposed linear model, the predicted values of DI [i.e., Eq. (28)] are compared with those measured in 11 groups of beams in bending (whose B, H, fc and the ﬁber aspect ratio are reported in the ﬁrst columns of Table 3). In a single group, at least two FRC beams, having the same geometrical and material properties and without steel reinforcing bars, are taken into consideration. The beams have different amounts of ﬁberreinforcement Vf (see column #7 of Table 3), here indicated in terms of volume fraction. The selected amounts of ﬁbers are limited to those that produce, unequivocally, the failure in tension. Indeed, when high ﬁber volume fractions are used, the strains detected in the compressed zones of a beam cross-section (Fig. 6c) can localize (Uchida et al. [11]). In such situations, the ascending branch of the Sargin's parabola depicted in Fig. 7 (ﬁb [4]), and used to obtain Eq. (28), is no longer representative of the FRC behavior. Furthermore, column #8 of Table 3 reports two letters that deﬁne, respectively, the geometry (S ¼ straight, H ¼ with hooked ends, F ¼ with ﬂat ends, U ¼ undulated) and the material (S ¼ steel, P ¼ plastic) of the ﬁbers used to reinforce the beams. The 11 groups considered in Table 3, in which the FRC beams are labeled with the original names given by the Authors, comprise both three point bending (Barros et al. [12], Fantilli et al. [10], Jones et al. [13], Lee and Barr [14]) and four point bending tests (Banthia and Gupta [15], Michels et al. [16], Mobasher et al. [17], Soetens and Matthys [18], Wu [19]). The values of Pcr* and Pu, experimentally measured for all the

147

beams, are collected in Table 4. According to Eq. (27), these loads provide the ductility index, which is reported in the same Table together with the values of Vf,min. The latter is evaluated for each group as indicated in Fig. 9c. The values of DI and of the normalized reinforcement ratio v ¼ Vf/Vf,min, both referred to a single beam, deﬁne a point in the diagram of Fig. 10b. In the same picture, the proposed linear function, as deﬁned by Eq. (28), is also reported and compared with the experimental data. Although the unavoidable dispersion of the results, the linear relationship between DI and v is signiﬁcantly conﬁrmed by the tests. Indeed, Fig. 10b shows a good agreement between the computation of DI [with Eq. (28)] and the experimental data collected in Tables 3 and 4. Moreover, if the non-dimensional parameters DI and v are introduced, the hypothesis used to calculate the pullout response (i.e., ﬁber symmetrically and orthogonally positioned with respect to the crack, Lf assumed as the characteristic length of the FRC in tension, and a single bond-slip relationship for all the types of the ﬁber) seem to be irrelevant to assess the brittle/ductile behavior of FRC beams. Speciﬁcally, this is particularly true for the ﬁber orientation, which affects in the same manner both the values of Pcr* and Pu in Eq. (27). For this reason, it does not inﬂuence the values of DI, as conﬁrmed by the tests performed on different FRC beams (see Table 4) and, therefore, with different ﬁber orientations. From a practical point of view, a simple-to-apply procedure, requiring the use of Eq. (28) and a single test on a full-scale specimen (i.e., a single FRC beam), can provide Vf,min. Indeed, from the ductility index measured in the test (i.e., DI1 in Fig. 10b), the corresponding value of the normalized reinforcement ratio v1 can be obtained through Eq. (28) (or graphically in Fig. 10b). Then the calculation of the minimum content of ﬁbers is possible by using the inverse formula Vf,min ¼ Vf/v1 (where Vf is the amount of ﬁbers in the tested beam). Thus, the proposed approach, which is a sort of compromise between (complex and not generalizable) theoretical methods and (onerous) experimental campaigns, introduces a more practical and user-friendly tool for the evaluation of the minimum content of ﬁbers, in order to have a deﬂection-hardening behavior of FRC beams.

5. Conclusions According to the analyses previously described, the following conclusions can be drawn:

Fig. 10. The computation of DI with the proposed relationship [Eq. (28)]: (a) comparison with the results of the general model (Tables 1 and 2); (b) comparison with the experimental data (Tables 3 and 4).

148

A.P. Fantilli et al. / Cement and Concrete Composites 65 (2016) 139e149

Table 4 Evaluation of the ductility index, and of the minimum amount of ﬁber, in the FRC beams tested in some experimental campaigns. Group

Beam

Pcr*(kN)

Pu (kN)

DI

Vf,min (%)

v

References

I

N2 N3 N4 F80/60_Cf10 F80/60_Cf20 F80/60_Cf30 Mix_1_1 Mix_1_2 Mix_1_3 Mix_2_1 Mix_2_2 Mix_2_3 Mix_3_1 Mix_3_2 Mix_3_3 50C(40) 50C(80) 75C(40) 75C(80) C25/30(25) C25/30(75) 0.20(A) 0.52(A) 0.65(A) 0.91(A) S13-HL-28d S26-HL-28d S39-HL-28d 4P-LN-20 4P-LN-40 DWP_0.2 DWP_0.5 DWP_1 DWP_1.5 FP_0.2 FP_0.5 FP_1 FP_1.5

25.0 27.5 21.1 14.4 15.9 13.6 18.2 18.6 20.6 21.3 22.5 23.1 22.8 21.5 20.4 2.3a 2.9a 6.5a 6.9a 13.8 16.2 25.8 28.2 29.0 31.6 23.3 26.1 25.1 3.6a 4.8a 12.0a

17.0 22.8 19.5 5.4 8.3 12.0 11.5 12.4 13.2 18.8 15.6 19.5 24.0 23.2 21.6 1.9a 3.6a 5.0a 8.9a 7.4 20.4 12.1 29.2 29.8 37.3 5.8 15.2 21.9 3.3a 5.9a 3.9 5.3 8.1 9.3 4.0 5.7 6.9 8.8

0.32 0.17 0.07 0.62 0.48 0.12 0.37 0.33 0.36 0.12 0.31 0.16 0.05 0.08 0.06 0.20 0.26 0.24 0.29 0.46 0.26 0.53 0.03 0.03 0.18 0.75 0.42 0.13 0.06 0.23 0.67 0.56 0.33 0.23 0.71 0.58 0.49 0.36

1.13

0.44 0.66 0.88 0.28 0.55 0.83 0.36

Banthia and Gupta [15]

II

III

IV V VI VII

VIII

IX X

XI

a

13.6a

0.46

1.02

Barros et al. [12]

Fantilli et al. [10]

0.72

1.08

0.73 0.74 0.73 0.64

0.56

0.31 2.05

2.89

0.70 1.39 0.69 1.38 0.44 1.31 0.31 0.81 1.01 1.42 0.30 0.59 0.89 0.83 1.66 0.10 0.24 0.49 0.73 0.07 0.17 0.35 0.52

Jones et al. [13]

Lee and Barr [14] Michels et al. [16]

Mobasher et al. [17]

Soetens and Matthys [18] Wu [19]

Value estimated from the experimental curves.

1. When the ultimate bending moment equates the effective cracking moment, which corresponds to the transition from the brittle to the ductile behavior, the minimum amount of ﬁbers (Vf,min) can be deﬁned for FRC beams. Such content of ﬁbers, having the same mechanical function of the minimum area of steel rebars in reinforced concrete beams, can also be obtained by imposing the ductility index DI equal to zero. 2. The values of DI linearly increase with the normalized reinforcement ratio v ¼ Vf/Vf,min, regardless of the geometrical and mechanical properties of materials. Both numerical approaches and experimental results seem to conﬁrm the existence of the proposed linear relationship [Eq. (28)]. 3. With respect to the current approaches, Eq. (28), accompanied by a single test on a full-scale FRC beam in bending, deﬁnes a more effective and user-friendly tool for the evaluation of Vf,min. Finally, further studies should be developed to introduce DI in FRC beams containing also steel rebars, in order to identify a possible relationship between the ductility index and the amount of reinforcement (ﬁbers and rebars) in hybrid structures.

Acknowledgment The grant given by the Italian Laboratories University Network of seismic engineering (ReLUIS), and used to develop this research work, is gratefully acknowledged.

References [1] A.P. Fantilli, H. Mihashi, T. Nishiwaki, Tailoring hybrid strain-hardening cementitious composites, ACI Mater. J. 111 (2) (2014) 211e218. [2] A.E. Naaman, Strain hardening and deﬂection hardening ﬁber reinforced cement composites, in: Proceedings of the International Workshop High Performance Fiber Reinforced Cement Composites, RILEM Publications, 2003, pp. 95e113. Pro. 30. [3] A.P. Fantilli, D. Ferretti, I. Iori, P. Vallini, Behaviour of R/C elements in bending and tension: the problem of minimum reinforcement ratio, Eur. Struct. Integr. Soc. 24 (1999) 99e125. de ration internationale du be ton, Model Code 2010-Final Draft, Volume [4] ﬁb - Fe 1, Switzerland, Lausanne, 2012 ﬁb Bulletin, 65. [5] A. Caratelli, A. Meda, Z. Rinaldi, Design according to MC2010 of a ﬁbrereinforced concrete tunnel in Monte Lirio, Panama, Struct. Concr. 13 (2012) 166e173. [6] A. de la Fuente, P. Pujadas, A. Blanco, A. Aguado, Experiences in Barcelona with the use of ﬁbres in segmental linings, Tunn. Undergr. Space Technol. 27 (2012) 60e71. [7] B. Chiaia, A.P. Fantilli, P. Vallini, Evaluation of crack width in FRC structures and application to tunnel linings, Mater. Struct. 42 (2009) 339e351. [8] A.P. Fantilli, P. Vallini, Bond-slip relationship for smooth steel reinforcement, in: N. Bicanic, R. de Borst, H. Mang, G. Meschke (Eds.), Computational Modelling of Concrete Structures (EURO-C 2003), St. Johann Im Pongau, March 17e20th, 2003, A.A. Balkema Publishers, Lisse, 2003, pp. 215e224. [9] B. Chiaia, A.P. Fantilli, P. Vallini, Evaluation of minimum reinforcement ratio in FRC members and application to tunnel linings, Mater. Struct. 40 (2007) 593e604. [10] A.P. Fantilli, A. Gorino, B. Chiaia, Precast plates made with lightweight ﬁberreinforced concrete, in: Proceedings of the FRC 2014 Joint ACI-ﬁb International Workshop, Montreal, July 24-25th, 2014, 2014, pp. 224e234. [11] Y. Uchida, M. Kawai, K. Rokugo, Back analysis of tensile stress-strain relationship of HPFRCC, in: Proceedings of the Fifth International RILEM Workshop on High Performance Fiber Reinforced Cement Composites, Mainz, July 10-13th, 2007, 2007, pp. 49e56.

A.P. Fantilli et al. / Cement and Concrete Composites 65 (2016) 139e149 [12] J.A.O. Barros, V.M.C.F. Cunha, A.F. Ribeiro, J.A.B. Antunes, Post-cracking behaviour of steel ﬁbre reinforced concrete, Mater. Struct. 38 (2005) 47e56. [13] P.A. Jones, S.A. Austin, P.J. Robins, Predicting the ﬂexural load-deﬂection response of steel ﬁbre reinforced concrete from strain, crack-width, ﬁbre pull-out and distribution data, Mater. Struct. 41 (2008) 449e463. [14] M.K. Lee, B.I.G. Barr, A four-exponential model to describe the behaviour of ﬁbre reinforced concrete, Mater. Struct. 37 (2004) 464e471. [15] N. Banthia, R. Gupta, Hybrid ﬁber reinforced concrete (HyFRC): ﬁber synergy in high strength matrices, Mater. Struct. 37 (2004) 707e716. [16] J. Michels, R. Christen, D. Waldmann, Experimental and numerical investigation on postcracking behavior of steel ﬁber reinforced concrete, Eng. Fract. Mech. 98 (2013) 326e349. [17] B. Mobasher, M. Bakhshi, C. Barsby, Backcalculation of residual tensile strength of regular and high performance ﬁber reinforced concrete from ﬂexural tests, Constr. Build. Mater. 70 (2014) 243e253. [18] T. Soetens, S. Matthys, Different methods to model the post-cracking behaviour of hooked-end steel ﬁbre reinforced concrete, Constr. Build. Mater. 73 (2014) 458e471. [19] Y. Wu, Flexural strength and behavior of polypropylene ﬁber reinforced concrete beams, J. Wuhan Univ. Technol. 17 (2) (2002) 54e57.

Nomenclature Ac: area of the cementitious matrix of an ideal tie (Fig. 2a) Af: area of the ﬁber cross-section B: width of a FRC beam cross-section DI: ductility index Ec: Young's modulus of concrete in tension and compression Ec1: secant modulus of elasticity from the origin of the stress-strain diagram to the peak of compressive stress of concrete Ef: Young's modulus of ﬁber F: axial load applied to a FRC tie (Fig. 1a) fc: compressive strength of concrete Fcr: cracking load of a FRC tie (Fig. 1a) fct: tensile strength of concrete Fu: axial load at the strain localization in a FRC tie (Fig. 1a) fu: tensile strength of ﬁber GF: fracture energy of concrete in tension H: depth of a FRC beam cross-section i: subscript referred to the generic cross-section of an ideal tie (Fig. 2a) j: subscript referred to the generic strip of a FRC beam cross-section (Fig. 6b) k: Ec / Ec1 ¼ plasticity number of FRC in compression [Eq. (25.a)] Lf: length of the ﬁber ltr: transfer length of an ideal tie (Fig. 2a) M: bending moment associated to the generic curvature m;

149

m: number of strips in a FRC beam cross-section Mcr: bending moment at ﬁrst cracking Mcr*: bending moment at the effective cracking Mu: bending moment at the strain localization N: axial load associated to the crack width w of an ideal tie (Fig. 2a) n: Ef /Ec ¼ ratio between the Young's moduli of the ﬁber and of the cementitious matrix P: load applied to a FRC beam in three point bending (Fig. 1b) Pcr: load corresponding to the ﬁrst cracking of a FRC beam in three point bending (Fig. 1b) Pcr*: load corresponding to the effective cracking of a FRC beam in three point bending (Fig. 1b) pf: perimeter of a ﬁber cross-section Pu: load corresponding to the strain localization of a FRC beam in three point bending (Fig. 1b) R: resultant of stresses in a FRC cross-section s, s1: slip between ﬁber and surrounding concrete, and the value at the peak in the bond-slip model [Eq. (13)] s0: value of s in the cracked cross-section of an ideal tie (Fig. 2b) v: normalized reinforcement ratio of a FRC beam Vf, Vf,min: ﬁber volume fraction of a FRC beam and its minimum value w, w1, wc: crack width, and limit values of w in the ﬁctitious crack model [Eq. (12)] y: vertical coordinate of a FRC cross-section (Fig. 6b) z: horizontal coordinate of an ideal tie (Fig. 2a) a: exponent in the bond-slip model [Eq. (13.a)] b: coefﬁcient in the bond-slip model [Eq. (13.b)] d: midspan deﬂection of a FRC beam in bending (Fig. 1b) Dh: H / m ¼ depth of the strips of a FRC beam cross-section (Fig. 6b) DL: elongation of a FRC tie (Fig. 1a) Dl: part of the transfer length εc1: FRC strain at the peak of compressive stress (Fig. 7) εF: compressive or tensile strain of FRC (Fig. 7) εF1: FRC strain at the peak of tensile stress (Fig. 7) h: j εF / εc1 j ¼ normalized strain of the FRC in compression [Eq. (25.a)] l: parameter of a FRC cross-sectional strain proﬁle m: curvature of a FRC cross-section sc: concrete stress of an ideal tie (Fig. 2c) sc,I: concrete stress in the Stage I cross-section of an ideal tie (Fig. 2c) sc0: concrete stress in the cracked cross-section of an ideal tie (Fig. 2c) sF: compressive or tensile stress of FRC (Fig. 7) sf: stress in the ﬁber of an ideal tie (Fig. 2c) sf,I: stress of the ﬁber in the Stage I cross-section of an ideal tie (Fig. 2c) sf0: stress of the ﬁber in the cracked cross-section of an ideal tie (Fig. 2c) t: bond stress tmax, tf ¼ maximum and residual values of t in the bond-slip model [Eq. (13)]: f: diameter of the ﬁber cross-section.

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