Determination of size-independent specific fracture energy of normal- and high-strength self-compacting concrete from wedge splitting tests

Construction and Building Materials 48 (2013) 548–553

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Determination of size-independent specific fracture energy of normal- and high-strength self-compacting concrete from wedge splitting tests Héctor Cifuentes a,⇑, Bhushan L. Karihaloo b a b

Grupo de Estructuras, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n, E41092 Seville, Spain University of Cardiff, School of Engineering, Cardiff CF24 3AA, UK

h i g h l i g h t s  Wedge splitting tests have been conducted on two different self-compacting concrete.  SCC showed a lower size-independent fracture energy than similar vibrated concrete.  The Young’s modulus of the self-compacting concrete mixes is higher.  The higher proportion of fine particles decrease ductility of SCC concrete mixes.

a r t i c l e

i n f o

Article history: Received 19 March 2013 Received in revised form 25 June 2013 Accepted 21 July 2013 Available online 14 August 2013 Keywords: Concrete Boundary effect Size-independent fracture energy Transition length Test conditions

a b s t r a c t Wedge splitting tests have been conducted on two different self-compacting concretes (normal- and high-strength) and the size-dependent fracture energy Gf determined from the measured RILEM work of fracture. Then the specific size-independent fracture energy, GF, has been determined using the boundary effect (BE) method of Hu and Wittmann and the simplified boundary effect (SBE) method proposed by Abdalla and Karihaloo. Tests on specimens of three different sizes and four different relative notch depths have shown that a unique value of GF can be obtained irrespective of the specimen size and relative notch depth. The results by both the BE and SBE methods are in very good agreement. A comparison with previous results from Abdalla and Karihaloo for normal- and high-strength vibrated concretes tested under the same conditions in the same laboratory shows that the SCC has a lower specific size-independent fracture energy than the vibrated concrete of the same strength. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Wedge splitting tests were developed by Linsbauer and Tschegg in 1986 [1] and subsequently modified by Bruhwiler and Wittmann in 1990 [2]. It is a very stable test for determining the fracture energy of concrete. The specimens used are very compact and require small amounts of material as compared to the notched beams employed in three-point bending tests. However, the implementation of this type of test requires more sophisticated tools than the three-point bending test and the number of results available in the literature obtained using the wedge splitting test of concrete is very limited. The specific fracture energy of concrete measured using the RILEM work-of-fracture procedure by both the wedge splitting and the three-point notched beam tests shows a size dependency [3]. ⇑ Corresponding author. Tel.: +34 954487485; fax: +34 954487295. E-mail address: [email protected] (H. Cifuentes). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.07.062

Hu and Wittmann [4] developed the boundary effect (BE) method that considers the influence of the back free boundary of the uncracked ligament area on the fracture process zone of concrete based on the concept of the local fracture energy. By means of this method it is possible to obtain a size-independent specific fracture energy of concrete based on the measured fracture energy of specimens of one size but different relative notch depths. Elices and coworkers [5–7] proposed an alternative method to obtain a sizeindependent specific fracture energy value, which consists of the identification of the sources of experimental error in the RILEM three-point bending method. They proposed a methodology for eliminating the major source of error, namely by including the work-of-fracture that is not measured in the RILEM method due to practical difficulties in capturing the tail part of the load–deflection plot. In a recent paper Cifuentes et al. [8], showed that if the size-dependent Gf is corrected following the methods of Elices and co-workers [5–7] and of Hu and Wittmann [4], then the resulting specific fracture energy GF is very nearly the same.

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Nomenclature GF WS BE SBE FPZ Gf gf x a W

a al

specific size-independent fracture energy wedge splitting boundary effect simplified boundary effect fracture process zone measured size-dependent fracture energy local fracture energy distance along the uncracked ligament initial notch depth depth of the specimen relative notch depth ligament transition length

Abdalla and Karihaloo [3] and Karihaloo et al. [9] extended and confirmed the boundary effect hypothesis of Hu and Wittmann [3] and observed that a size-independent specific fracture energy GF of concrete could be obtained by testing three point bend (TPB) or wedge splitting (WS) specimens containing either a very shallow or a deep starter notch. This observation was based on TPB and WS tests, and they proposed the simplified boundary effect (SBE) method [9] that greatly reduces the number of specimens to be tested. Abdalla and Karihaloo [3] carried out a comprehensive experimental study of normal- and high-strength vibrated concrete using the WS test for three different specimen sizes (W = 100, 200 and 300 mm) and four different relative notch depths (a = a/W = 0.2, 0.3, 0.4 and 0.5) [3]. It is the aim of the present paper to show the results of a new set of wedge splitting tests carried out on normal- and highstrength self-compacting concretes and to compare the fracture behavior of these self-compacting concretes with that of vibrated normal- and high-strength concrete made and tested in the same laboratory and under the same test conditions. Ninety-six notched specimens were tested according to the WS test method to determine the size-dependent fracture energy from the RILEM work-of-fracture. The geometry of the specimens was the same as employed by Abdalla and Karihaloo [3] for vibrated concrete. The BE and SBE methods were applied to obtain an estimate of the size-independent fracture energy and the ligament transition length of concrete. The results show a good agreement between the BE and the SBE methods for the size-independent specific fracture energy and the ligament transition length of selfcompacting concrete. A comparison with previous results for normal- and high-strength vibrated concretes tested under the same conditions in the same laboratory shows that the SCC has a lower specific size-independent fracture energy than the vibrated concrete of the same strength.

CMOD F P h h CMODc COV fc fst Ec lch

crack mouth opening displacement vertical force at bearings horizontal force at bearings half-angle of the wedge height of the groove on top of wedge splitting specimen crack mouth opening displacement at the end of test coefficient of variation compressive strength splitting tensile strength Young’s modulus characteristic length

ligament size al is a parameter depending on both the material properties and the specimen geometry. On the basis of the BE method of Hu and Wittmann [4] the sizedependent measured fracture energy, Gf, represents the average value of the variable local fracture energy, gf, which depends on the distance, x, along the un-cracked ligament length (Fig. 1)

Gf ðaÞ ¼

1 W a

Z

Wa

g f ðxÞdx

ð1Þ

0

Substituting the bi-linear approximation for the local fracture energy variation (Fig. 1) into Eq. (1) and introducing the dimensionless ratios a = a/W and al = al/W, a relation between the measured fracture energy, Gf, the transition length, al and the size-independent fracture energy, GF is obtained

( Gf ðaÞ ¼

 GF 1  12

al

1a a GF 12 1 al



1  a > al 1  a  al

ð2Þ

Making use of Eq. (2), the size-independent fracture energy of concrete and the transition length can be back-calculated from the size-dependent fracture energy Gf(a). To do that, it is necessary to get the size-dependent fracture energy of a specimen of a given size with a full range variation of the relative notch depth a. Usually, the number of the measured Gf(a) values is therefore much larger than the two unknowns GF and al in Eq. (2). For this reason, the over-determined system of equations is solved by a least squares method to obtain the best estimates of GF and al. Duan [14] showed that although the measured values Gf(a) depend on a (and of course on the specimen size), the above procedure indeed leads to a GF value that is essentially independent of the specimen size and relative notch depth [13].

2. Theoretical background Hu and Wittmann [4,10–13] argued that the effect of the free boundary is felt in the fracture process zone (FPZ) ahead of a real crack so that the energy required to create a fresh crack decreases as the crack approaches the free boundary. Initially, when the crack growths from a pre-existing notch, the rate of decrease is moderate, almost a constant, but it accelerates as the crack approaches the end of the un-cracked ligament. They represented the transition from the moderate decrease to the rapid decrease by a bi-linear approximation (Fig. 1). The bi-linear function consists of a horizontal line with the value of GF and a descending branch that reduces to zero at the back surface of the specimen [10]. The intersection of these two straight lines is defined as the transition ligament size al or the crack reference length [11]. The transition

Fig. 1. Bi-linear approximation of the local fracture energy of concrete according to Hu [10].

550

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Based on the BE method of Hu and Wittmann, Abdalla and Karihaloo [3] and Karihaloo et al. [9] have proposed and validated a simplified method (SBE) that allows the estimation of the sizeindependent fracture energy by testing specimens of a single size with only two relative notch depths, one the shallowest possible and the second the deepest possible. In this manner, there are only two Eq. (2) to be solved in two unknowns GF and al. This simplified method eliminates the need for the least squares method for the solution of an over-determined system of simultaneous equations and the time consuming testing of a large number of specimens with different W and a/W. They have demonstrated that the necessary values of the smallest relative notch depth must be a/W 6 0.1 for three-point bending and 60.2 for wedge splitting tests. The values of the deepest starter notch must be a/W P 0.5 for both the TPB and WS tests [9]. In recent papers Muralidhara et al. [15,16] and Karihaloo et al. [17] have revealed that the local fracture energy distribution along the unbroken ligament can also be approximated by a tri-linear curve. This tri-linear model is closer to how the local fracture energy varies during crack growth in a notched concrete specimen, as evidenced by acoustic emission data [15]. However, Karihaloo et al. [17] showed that the bi-linear model of Hu and Wittmann [4] and the simplified version of Abdalla and Karihaloo [3] is quite adequate in practice to determine the size-independent fracture energy of concrete mixes.

BS1881:part116:1983. The indirect tensile strength (fst) was obtained using the cylinder splitting test according to BS1881:part117:1983. The splitting tests were carried out on 100 mm diameter by 200 mm long cylinders. The static elastic modulus of concrete (Ec) was obtained according to the BS1881:part1 21:1983 by gradually loading a cylindrical specimen in compression to approximately a third of its failure load and measuring the corresponding strain. The strain was measured using 30 mm strain gauges. The measured mechanical properties (mean values and coefficient of variation) of the concretes are given in Table 2. Ninety-six cubical test specimens of different sizes (100, 200 and 300 mm) as shown in Fig. 2a were tested in wedge splitting mode. The notch to depth ratios a/W were 0.2, 0.3, 0.4 and 0.5. The testing was carried out using a Dartec closed-loop testing machine (200 kN). The rate of loading was controlled by a crack mouth opening displacement (CMOD) gauge at a very low rate (0.0002 mm/s) so that the fracture occurred in a stable manner. The loading arrangement for the WS test is shown in Fig. 2a. The notches were made in the groove using diamond saw with a blade of 3 mm in thickness and over 300 mm in diameter. In the wedge splitting test the specific fracture energy of concrete is given by

3. Experimental procedure

where CMODc is the value of CMOD at the end of the test and B = W. The definition of the notch size and depth is shown in Fig. 2b and the values for different specimen sizes are given in Table 3. The splitting load P is calculated from

The tests described in this paper were conducted on normal and high-strength self-compacting concretes in the same way and in the same laboratory as the previous tests carried out by Abdalla and Karihaloo [3] on similar strength vibrated concretes. The mixes of the self-compacting concrete made were designed according to the method proposed by Deeb and Karihaloo [18]. Their nominal cube compressive strengths were 40 and 100 MPa, respectively. The mix proportions and constituents are showed in Table 1. The maximum aggregate size was 10 mm and the cement was an ordinary Portland cement CEMII/B-V 32.5R. The mixes were prepared in a planetary mixer by mixing the coarsest constituent (coarse aggregate) and the finest one (microsilica), followed by the next coarsest (sand) and next finest constituent (cement), and so on. Before each addition, the constituents were mixed for 2 min. To fluidize the dry mix, two-thirds of the super-plasticizer (SP) was added to the water. One-half of this water–SP mixture was added to the dry constituents and mixed for 2 min. One-half of the remaining water–SP mixture was then added and mixed for 2 min. This process was continued until all water–SP mixture was added in about 10 min. The remaining one-third of the SP was added and mixed for 2 min just before transferring the mix into the moulds. The characteristic compressive strength (fc) was determined from the crushing of 100 mm cubes in accordance with Table 1 Constituents and proportions for self-compacting concrete mixes (kg/m3). Constituents

NSSCC

HSSCC

Cement Micro-silica Coarse aggregates (crushed limestone) <10 mm Sand <2 mm Water GGBS Limestone powder (<2 mm) Super-plasticizer/cement Water/binder Flow spread (mm) T500 (s)

263 0 859 701 180 112 170 0.6% 0.48 750 2.7

460 69 911 607 123 0 97 3.2% 0.23 710 3.0

R CMODc Gf ða; WÞ ¼

P¼

PdCMOD BðW  h  aÞ

0

ð3Þ

F 2 tan h

ð4Þ

where F is the vertical force on the bearings (assuming the frictional contribution to be negligible) and h is one-half of the wedge angle (Fig. 2c). 4. Results Typical recorded load–CMOD diagrams are shown in Fig. 3 from which the fracture energy Gf(a,W) was calculated using Eq. (3). Table 4 shows the results of the measured fracture energy, Gf(a,W), with an indication of the mean value and the coefficient of variation (COV,%). All values show a size-dependent trend in according with that observed by other authors; it decreases when the ligament size increases (i.e. when either the size of the specimen increases or the relative notch depth decreases). The variation of the Gf(a,W) with specimen size is less pronounced for the high-strength concrete due to the more brittle behavior of this concrete compared to the normal-strength concrete. This effect can be appreciated by comparison of the ultimate values of CMOD for NSSCC and HSSCC in Fig. 3. The test results for Gf(a,W) from the WS tests were substituted into Eq. (2) in order to determine the size-independent fracture energy, GF, and the transition ligament length, al, according to the BE method. As the number of results of Gf(a,W) for each depth, W, and the full range of the relative notch depth, a, was 4 (more than the number of unknowns), the system of equations was solved by a least squares method to get the best estimate of GF and al. These Table 2 Mechanical properties of concrete mixes. Concrete

fc (MPa)

fst (MPa)

Ec (GPa)

NSSCC HSSCC

41.0 ± 11% 97.7 ± 9%

3.9 ± 9% 5.5 ± 5%

32.8 ± 9% 45.3 ± 7%

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Fig. 2. Wedge splitting test geometry and dimensions.

Table 3 Specimen dimensions and the relative notch depth a. W

a

a0

a = a0/(W  dn)

h

dn

f

T

h (°)

100

12 20.5 29 37.5

17 25.5 34 42.5

0.2 0.3 0.4 0.5

15

20

30

50

14.5

30 47.5 65 82.5

35 52.5 70 87.5

0.2 0.3 0.4 0.5

25

41 67.5 94 120.5

53 79.5 106 132.5

0.2 0.3 0.4 0.5

35

200

300

30

47

60

90

100

150

15.0

15.5

values are shown in Table 5. The simplified method (SBE) proposed by Abdalla and Karihaloo [3] to obtain the size-independent fracture energy from the Gf(a,W) values corresponding to the deepest (0.5) and shallowest (0.2) relative notch depths were also obtained and are also shown in Table 5. The results in Table 5 show that the size-independent specific fracture energy GF remains constant for the three different speci-

men sizes investigated here. The accuracy between results from BE and SBE methods is very high even for the ligament transition length values. The ligament transition length al for both types of concrete shows the same trend; it increases with the specimen size. 5. Comparison with results obtained by Abdalla and Karihaloo for vibrated concrete Fig. 4 shows the size-independent specific fracture energy obtained by applying Eq. (2) to the measured size-dependent specific fracture energy Gf(a,W) from WS tests for normal- and highstrength self-compacting concrete and compares it with the values obtained by Abdalla and Karihaloo [3] for vibrated concrete. The vibrated concrete mixes tested by Abdalla and Karihaloo in the same laboratory and under the same conditions were normal- and highstrength concrete with a compressive strength of 59.3 and 98.4 MPa respectively. The size-independent fracture energies for the vibrated mixes were 155 and 123 N/m, their split tensile cylinder strengths were 4.3 and 6.2 MPa and the Young’s moduli were 38.3 and 43.0 GPa, respectively [19]. A comparison of HSSCC and HSC mixes with a very similar compressive strength shows that the fracture energy of self-compacting

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Fig. 3. Recorded load–CMOD diagrams for NSSCC (a) and HSSCC (b).

concrete is lower than the corresponding vibrated concrete. This can be explained by the fact that the HSSCC concrete contains less coarse aggregate and thus a shorter frictional part of the load– CMOD diagram. The difference is less noticeable in normal strength concretes. Although the compressive strength of the NSSCC mix (C40) is lower than the NSC vibrated mix (C60), the specific fracture energy of these mixes is quite similar. In order to perform a more detailed analysis of the differences in the fracture behaviour between the self-compacting and vibrated mixes, the ductility of concrete is analyzed by means of the characteristic length [17]:

lch ¼

Ec GF fct2

ð5Þ

where fct is the direct tensile strength of concrete and considered as fct = 0.65fst [20]. Table 6 shows the values of the characteristic length for the vibrated and self-compacting concrete mixes. A comparison of lch for HSSCC and HSC mixes shows that the self-compacting concrete is more brittle than a similar strength Table 4 Size dependent fracture energy Gf(a,W). Concrete

NSSCC

HSSCC

a

Fig. 4. Values of the size-independent fracture energy for NSSCC, NSC, HSSCC and HSC.

Table 5 Estimated values of GF and al with the BE and SBE methods applied to one-size specimens with different relative notch depths. Concrete

Parameter

Method

W (mm) 100

200

300

0.2 0.3 0.4 0.5

87.2 ± 9% 80.3 ± 4% 66.5 ± 7% 59.0 ± 4%

108.8 ± 2% 102.8 ± 1% 108.8 ± 2% 83.0 ± 13%

111.6 ± 15% 105.2 ± 19% 93.0 ± 17% 86.6 ± 15%

0.2 0.3 0.4 0.5

68.4 ± 14% 64.6 ± 15% 61.7 ± 11% 57.6 ± 5%

70.5 ± 4% 67.6 ± 13% 66.1 ± 5% 60.6 ± 3%

73.2 ± 11% 70.8 ± 13% 65.3 ± 16% 63.8 ± 9%

NSSCC

HSSCC

W (mm) 100

200

300

GF (N/m)

BE SBE

148.9 136.6

152.8 151.6

153.6 153.2

al (mm)

BE SBE

65.3 57.9

92.0 90.5

134.7 130.5

GF (N/m)

BE SBE

90.6 90.7

86.3 87.0

89.1 88.9

al (mm)

BE SBE

39.2 39.4

58.7 60.7

88.4 84.6

H. Cifuentes, B.L. Karihaloo / Construction and Building Materials 48 (2013) 548–553 Table 6 Characteristic length of self-compacting and vibrated [16] concrete mixes. Concrete

lch (mm)

NSSCC (C-40) HSSCC (C-100) NSC (C-60) HSC (C-100)

796 310 757 330

vibrated concrete. As expected, the NSSCC (C40) concrete mix is more ductile than the higher strength (C60) vibrated mix, despite the fact that their specific fracture energies are very similar. 6. Discussion of results The results provided in this work contribute to increase of experimental data available for WS tests for self-compacting concrete mixes. In the literature the number of experimental results by using wedge splitting test is very limited, especially in case of self-compacting concrete. Since the tests were conducted under similar conditions to those carried out by Abdalla and Karihaloo [3] for vibrated concrete, the following similarities and differences were observed:  The fracture energy of normal- and high-strength self-compacting concrete measured using the RILEM work-of-fracture method showed a size dependency with a trend usual for this type of test (Table 4) and similar to that observed in vibrated concrete. The measured specific fracture energy increases with the size of the ligament area (i.e. when the specimen depth increases or the relative notch depth decreases).  By using the size-dependent values of the measured fracture energy and applying Eq. (2) for one-size specimens with the full range of relative notch depths a unique value of the size-independent fracture energy of each concrete was obtained according to BEM (Table 5). The application of the simplified boundary effect method showed very similar results thus confirming the simplified BEM proposed by Abdallla and Karihaloo [3]. The values of the obtained GF were almost constant for each type of concrete irrespective of the test specimen size.  A comparison of GF of HSSCC and HSC mixes with a very similar compressive strength showed a significant decrease in the facture energy of the self-compacting mix. This reduction is due to the lower content of coarse aggregates in the self-compacting concrete as these contribute to a higher values of the fracture energy through a more pronounced tail part of the load–CMOD diagram [21].  A better understanding of the ductility/brittleness is therefore provided by the characteristic length of the mix. A comparison of lch for HSSCC and HSC mixes shows that the self-compacting concrete is more brittle than a similar strength vibrated concrete. However as expected, the NSSCC (C40) concrete mix is somewhat more ductile than the higher strength (C60) vibrated mix, despite the fact that their specific fracture energies are very similar. Lower strength concretes are in general more ductile than higher strength concrete mixes. Thus, NSSCC (C40) would be expected to be more ductile than NSC (C60). The reason that is not so is because the self-compacting concrete mixes are inherently more brittle. 7. Conclusions From the analysis of the wedge splitting results obtained for normal- and high-strength self-compacting concrete and by comparison with the available results previously obtained in the same

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laboratory and under the same conditions for normal- and highstrength vibrated concrete, the following conclusions can be drawn: – The specific fracture energy of self-compacting concrete mixes is lower than that of vibrated mixes of the same strength. This is due to the lower content of coarse aggregate in the self-compacting concrete. – The ductility of vibrated concrete mixes as measured by their characteristic length is only marginally higher than that of self-compacting concrete mixes. This is due to the fact that although their specific fracture energy is higher, as mentioned above, the internal stiffness of the matrix and consequently the Young’s modulus of the self-compacting concrete mixes is higher and the tensile strength lower because they contain a higher content of fine particles.

Acknowledgements HC would like to acknowledge financial support provided for this research by the Spanish Ministry of Science and Technology under project BIA2010-21399-C02–02. This work was completed during HC’s sabbatical in Cardiff University. References [1] Linsbauer HN, Tschegg EK. Fracture energy determination of concrete with cube-shaped specimens. Zement und Beton 1986;31:38–40. [2] Bruhwiler E, Wittmann FH. The wedge splitting test: a method of performing stable fracture mechanics tests. Eng Fract Mech 1990;35(1–3):117–25. [3] Abdalla HM, Karihaloo BL. Determination of size-independent specific fracture energy of concrete from three-point bend and wedge splitting tests. Mag Concr Res 2003;55(2):133–41. [4] Hu X, Wittmann F. Fracture energy and fracture process zone. Mater Struct 1992;25:319–26. [5] Guinea GV, Planas J, Elices M. Measurement of the fracture energy using threepoint bend tests: Part 1—influence of experimental procedures. Mater Struct 1992;25(4):212–8. [6] Planas J, Elices M, Guinea GV. Measurement of the fracture energy using threepoint bend tests: Part 2 – influence of bulk energy dissipation. Mater Struct 1992;25:305–12. [7] Elices M, Guinea GV, Planas J. Measurement of the fracture energy using threepoint bend tests: Part 3 – Influence of cutting the P-d tail. Mater Struct 1992;25:327–34. [8] Cifuentes H, Alcalde M, Medina F. Measuring the size-independent fracture energy of concrete. Strain 2013;49(1):54–9. [9] Karihaloo BL, Abdalla HM, Imjai T. A simple method for determining the true specific fracture energy of concrete. Mag Concr Res 2003;55(5):471–81. [10] Hu X. Influence of fracture process zone height on fracture energy of concrete. Cem Concr Res 2004;34(8):1321–30. [11] Hu X, Duan K. Size effect: influence of proximity of fracture process zone to specimen boundary. Eng Fract Mech 2007;74(7):1093–100. [12] Hu X, Wittmann F. Size effect on toughness induced by crack close to free surface. Eng Fract Mech 2000;65:209–21. [13] Duan K, Hu X, Wittmann F. Size effect on specific fracture energy of concrete. Eng Fract Mech 2007;74(1–2):87–96. [14] Duan K. Boundary effect on concrete fracture and non-constant fracture energy distribution. Eng Fract Mech 2003;70(16):2257–68. [15] Muralidhara S, Prasad BKR, Eskandari H, Karihaloo BL. Fracture process zone size and true fracture energy of concrete using acoustic emission. Constr Build Mater 2010;24(4):479–86. [16] Muralidhara S, Raghu Prasad BK, Karihaloo BL, Singh RK. Size-independent fracture energy in plain concrete beams using tri-linear model. Constr Build Mater 2011;25(7):3051–8. [17] Karihaloo BL, Murthy AR, Iyer NR. Determination of size-independent specific fracture energy of concrete mixes by the tri-linear model. Cem Concr Res 2013;49:82–8. [18] Karihaloo BL, Ghanbari A. Mix proportioning of self-compacting high- and ultra-high-performance concretes with and without steel fibres. Mag Concr Res 2012;64(12):1089–100. [19] Karihaloo BL, Abdalla HM, Xiao QZ. Deterministic size effect in the strength of cracked concrete structures. Cem Concr Res 2006;36(1):171–88. [20] Neville AM. Properties of concrete. 4th ed. Prentice Hall; 1996. [21] Karihaloo BL. Fracture mechanics and structural concrete. USA: Longman Scientific and Technical Publishers; 1995.