Influence of mix composition and strength on the fracture properties of self-compacting concrete

Construction and Building Materials 110 (2016) 312–322

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Influence of mix composition and strength on the fracture properties of self-compacting concrete W.S. Alyhya 1, M.S. Abo Dhaheer 2, M.M. Al-Rubaye 3, B.L. Karihaloo ⇑ School of Engineering, Cardiff University, Cardiff CF24 3AA, UK

h i g h l i g h t s
Role of composition parameters of SCC mixes on their fracture behaviour examined.
Specific fracture energy increases as coarse aggregate volume fraction increases.
Increase in paste to solids ratio results in large decrease in fracture energy.
Increase in water to binder ratio reduces specific fracture energy.
Critical crack opening is dominated by the coarse aggregate volume.

a r t i c l e

i n f o

Article history: Received 11 June 2015 Received in revised form 29 January 2016 Accepted 15 February 2016

Keywords: Self-compacting concrete (SCC) Coarse aggregate volume Paste volume Notch depth Size-independent fracture energy Bilinear tension softening diagram

a b s t r a c t Self-compacting concrete (SCC) has undergone extensive investigations that have led to confidence in its fresh and hardened properties, yet its composition variations raise concerns as to its fracture behaviour. This paper presents the results of an experimental study on fracture behaviour of SCC mixes differing by the coarse aggregate volume, paste to solids ratio (p/s) and water to binder (or cementitious material (w/cm)) ratio. First the size-dependent fracture energy (Gf ) has been determined using the RILEM workof-fracture test on three point bend specimens of a single size, half of which contained a shallow starter notch (notch to depth ratio = 0.1), while the other half contained a deep notch (notch to depth ratio = 0.6). Then the specific size-independent fracture energy (GF ) was calculated using the simplified boundary effect formalism in which the variation in the fracture energy along the unbroken specimen ligament is approximated by a bilinear diagram. Finally, the bilinear approximation of the tension softening diagram corresponding to GF has been obtained using the non-linear hinge model. It is found that GF and the critical crack opening (wc) are dominated by the coarse aggregate volume in the mix and the mix grade. The larger the coarse aggregate volume (or the smaller the paste to solids ratio) the larger are both the mix toughness ðGF Þ and the critical crack opening (wc). However, the higher the mix grade the larger is the mix toughness ðGF Þ but the lower is the critical crack opening (wc). Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Specific fracture energy and the tension softening diagram of a concrete mix are the most important parameters describing its fracture behaviour. They form a basis for the evaluation of the load carrying capacity of cracked concrete structures [1,2]. According to RILEM recommendations [3], the specific fracture energy (or toughness) can be obtained by the work-of-fracture method

⇑ Corresponding author. 1 2 3

E-mail address: [email protected] (B.L. Karihaloo). On leave from University of Karbala, Iraq. On leave from University of Al-Qadisiyah, Iraq. On leave from University of Babylon, Iraq.

http://dx.doi.org/10.1016/j.conbuildmat.2016.02.037 0950-0618/Ó 2016 Elsevier Ltd. All rights reserved.

requiring tests on notched three-point bend specimens of different sizes and notch to depth ratios. It is however widely recognised [4–10], that the specific fracture energy of concrete obtained using the RILEM method is dependent on the size of the test specimen and the notch to depth ratio. To eliminate this size dependency, Guinea and co-workers [11–13] and Hu and Wittmann [14] proposed methods to correct the measured size-dependent specific fracture energy (Gf ) in order to obtain a size-independent value (GF ). The methodology proposed by Guinea and co-workers [11–13] involves adding the non-measured work-of-fracture due to the curtailment of the tail of the load–central deflection (P–d) curve recorded in the threepoint bend test. On the other hand, the methodology of Hu and Wittmann [14] is based on the observation that the local specific

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(Shallow notch)

(Deep notch)

Fig. 1. Schematic representation of the three-point bending test.

Fig. 2. Bilinear local fracture energy Gf ðWa Þ variation along the un-notched ligament of a notched specimen.

energy along the initially un-cracked specimen ligament varies during the crack propagation, the variation becoming more pronounced as the crack approaches the stress-free back face of the specimen, the so-called free boundary effect. Abdalla and Karihaloo [4] and Karihaloo et al. [15] simplified the free boundary effect formalism of Hu and Wittmann [14]. They proposed and validated extensively a simplified method by which the size-independent fracture energy can be determined by testing only geometrically identical specimens of the same size, half of which contain a shallow starter notch (notch to depth ratio = 0.1), while the other half contain a deep notch (notch to depth ratio = 0.6). Their method significantly reduces the number of specimens to be tested and eliminates the need for using the least squares method to solve an overdetermined system of simultaneous equations, as required in the Hu and Wittmann [14] method. Besides the size-independent fracture energy ðGF Þ, the analysis of cracked concrete structures using the non-linear fictitious crack model [16] requires the tension softening diagram r(w) of the

concrete mix relating the residual stress transfer capability r to the opening displacement w of the fictitious crack faces. As the determination of the tension softening diagram using the direct tension test is not a simple task [1], it is often approximated by a bilinear relationship whose parameters are determined in an inverse manner by matching the experimental load–displacement curve of a notched three-point bend beam. For this an analytical model based on the concept of a non-linear hinge was proposed by Ulfkjaer et al. [17] and Olesen [18]. In this model, the flexural response of a notched beam is obtained by allowing the fictitious crack to develop from the pre-existing notch in the central region of the beam where the bending moment is the largest. The width of this region, proportional to the beam depth, fixes the width of the non-linear hinge. Outside of this region, the material is assumed to behave in a linear elastic manner. Abdalla and Karihaloo [19], and Murthy et al. [20] showed how the non-linear hinge model can be adapted to construct the bilinear tension softening diagram of a concrete mix corresponding to its size-independent specific fracture energy. The fracture behaviour of concrete is significantly influenced by the properties of the interfacial transition zone (ITZ) [21], which in turn are governed by the mix ingredients in vibrated concrete (VC) and self-compacting concrete (SCC), as well. In comparison with VC, SCC requires relatively high amounts of fine particles and paste, but low coarse aggregate content [22–25]. Although SCC has passed from the research phase into real application, the differences in its composition from VC raise concerns about its fracture behaviour [26,27]. The concern is primarily because a lower coarse aggregate content in an SCC mix relative to a VC mix of the same grade is likely to reduce its energy absorption capacity and thus its ductility. This needs to be addressed. Previous work [28–33] on this topic was based on the size-dependent specific fracture energy, apart from the work of Cifuentes and Karihaloo [32] who

Fig. 3. Trilinear approximation of local fracture energy gf variation over the un-notched ligament length.

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Table 1 Mix proportions of test SCC mixes, kg/m3. cma

Mix designation

30A* 30B* 30C* 60A 60B 60C 80A 80B 80C

FAd b

Cement

ggbs

Water

SP

240 240 240 315 315 315 367.5 367.5 367.5

80 80 80 105 105 105 122.5 122.5 122.5

201.6 201.6 201.6 197.5 197.5 197.5 171.5 171.5 171.5

1.1 1.6 2.3 2.0 2.4 2.8 2.8 3.0 3.5

c

w/cm

SP/cm

LP

0.63 0.63 0.63 0.47 0.47 0.47 0.35 0.35 0.35

0.44 0.50 0.72 0.48 0.57 0.67 0.57 0.61 0.80

109 156 194 94 125 172 94 125 172

⁄⁄

CAe

p/s by vol.

924 840 756 924 840 756 924 840 756

0.61 0.67 0.72 0.69 0.72 0.79 0.69 0.72 0.79

⁄⁄⁄

FA

FA

164 234 291 141 188 258 141 188 258

579 530 504 536 528 477 536 529 478

*

A, B and C refer to the decrease in coarse aggregate and increase in paste volume for the same strength grade. Cementitious material, i.e. binder. b Super-plasticizer. c Limestone powder <125 lm. d Fine aggregate <2 mm (Note: a part of the fine aggregate is the coarser fraction of the limestone powder, FA⁄⁄ 125 lm–2 mm, whereas FA⁄⁄⁄ refers to natural river sand <2 mm). e Coarse aggregate <20 mm. a

Table 2 Flow and passing ability test results of SCC mixes. Mix designation

Slump flow

30A 30B 30C 60A 60B 60C 80A 80B 80C

V-funnel

J-ring flow test

Spread mm

t500 s

tv-funnel s

Spread mm

t500J s

t200 s

t400 s

H2/H1

685 665 655 665 650 655 730 750 670

0.50 0.88 0.81 1.18 1.32 1.40 1.92 2.06 2.09

2.46 2.47 2.76 3.23 3.54 4.04 6.67 7.34 7.44

665 635 650 640 645 630 705 730 655

0.60 1.04 0.74 1.48 1.43 1.60 2.43 2.70 2.80

0.47 0.57 0.53 0.77 0.81 0.81 1.45 1.62 1.45

1.08 1.11 1.10 1.48 1.72 1.65 3.10 3.20 3.07

0.91 0.84 0.92 0.89 0.84 0.87 0.93 0.90 0.91

used the model of Hu and Wittmann [14] and its simplified version proposed by Karihaloo et al. [15]. It is the aim of the present paper to investigate in detail the role of several composition parameters of SCC mixes in their fracture behaviour. In particular, the influence of coarse aggregate volume, paste to solids (p/s) and water to binder (w/cm) ratios on the sizeindependent fracture energy (GF Þ will be studied using the simplified boundary effect approach (SBE) suggested by Abdalla and Karihaloo [4] and validated by Karihaloo et al. [15]. The corresponding bilinear approximation of the tension softening diagram will then be obtained using the procedure based on the non-linear hinge model proposed by Abdalla and Karihaloo [19], and Murthy et al. [20].

2. Theoretical background The specific fracture energy (Gf Þ, as defined by RILEM technical committee, is the average energy given by dividing the total work of fracture by the projected fracture area (i.e. cross-section of initially un-cracked ligament) based on the load–displacement P–d curve. Hence, for a specimen of depth W, thickness B and initial notch depth a (as schematically shown in Fig. 1) the specific fracture energy ðGF Þ can be expressed as:

Gf ¼

1 ðW  aÞB

Z Pdd

ð1Þ

The specimen weight can be neglected for the small specimens used in this study.

L-box test

The specific fracture energy (Gf Þ can also be determined using a local energy gf concept described by Duan et al. [34,35] as follows (see Fig. 2):

Gf

a W

¼

1 W a

Z

Wa

g f ðxÞdx

ð2Þ

0

Hu and Wittman [14] proposed a bilinear approximation for the local fracture energy variation (g f ) along the crack path (Fig. 2) with the intersection of the two asymptotes defining a transition ligament size (al ). The latter, unlike the asymptotic value of specific fracture energy ðGF Þ, varies with the material properties and specimen geometry. A relation between the measured size-dependent fracture energy (Gf ), the transition length (al ) and the size-independent fracture energy (GF Þ can be obtained by substituting the bilinear approximation for the local fracture energy variation (Fig. 2) into Eq. (2)

8   a > al > GF 1  Wl 1  Wa > W a < a > 2ð1W Þ   Gf ¼ > W ð1Wa Þ al > > 1  Wa 6 W al : GF

ð3Þ

2W

The values of GF and al of a concrete mix are obtained once the mean size-dependent specific fracture energy (Gf ) of the mix has been measured on specimens of identical sizes, half of which have a shallow starter notch (a/W = 0.1), while the other half have a deep starter notch (a/W = 0.6) by the RILEM workof-fracture method using Eq. (1). Hu and Duan [36] showed that although the measured values of Gf depend on W and a/W, the

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5.0 a/W=0.1 a/W=0.6

Load, kN

4.0

3.0

2.0

1.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Deflection, mm 7.0 a/W=0.1 6.0

a/W=0.6

Load, kN

5.0 4.0 3.0 2.0 1.0 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Deflection, mm 10.0 a/W=0.1 a/W=0.6

Load, kN

8.0

6.0

4.0

2.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Deflection, mm Fig. 4. Typical load–displacement diagrams of two notched beams (out of twelve tested) from SCC mixes: 30A (top), 60B (middle), and 80C (bottom).

above procedure indeed leads to a GF value that is essentially independent of the specimen size and relative notch depth. In recent works, a trilinear approximation of the local fracture energy along the unbroken ligament was proposed by Muralidhara et al. [37,38], and Karihaloo et al. [39]. As has been evidenced by acoustic emission data, the trilinear approximation is closer to how the local fracture energy varies as the

crack grows from a notched specimen [37]. The local fracture energy (Gf Þ first rises from the fictitious boundary (notch tip), then remains nearly constant GF , before reducing again as the crack approaches both the stress-free back face boundary, Fig. 3. The Gf and GF relationship for the trilinear approximation is given in Eq. (4):

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Table 3 Measured fracture energy, Gf (a, W) for different SCC mixes from three point bending test (TPB). Mix designation

W mm

a/W

Mean [St. dev.] Gf (a, W), N/m

COV %

30A

100

30B

100

30C

100

60A

100

60B

100

60C

100

80A

100

80B

100

80C

100

0.1 0.6 0.1 0.6 0.1 0.6 0.1 0.6 0.1 0.6 0.1 0.6 0.1 0.6 0.1 0.6 0.1 0.6

96.20 53.50 85.90 53.00 73.40 52.30 108.6 65.80 91.90 56.50 83.90 51.90 105.5 58.50 100.1 57.00 97.60 57.70

9.20 9.30 9.00 7.60 10.0 8.20 10.7 2.60 6.20 8.85 11.4 6.10 5.30 9.80 9.90 8.60 11.3 4.30

[8.90] [5.00] [7.70] [4.00] [7.30] [4.30] [11.6] [1.70] [5.70] [5.00] [9.60] [3.20] [5.50] [5.70] [9.90] [4.90] [11.0] [2.50]

Table 4 Results of f cu , f st , E, GF and al of test SCC mixes.

notch to depth ratios is first determined by the RILEM method. Then Eq. (4) is applied to the mean values of Gf different notch to depth ratios. This gives an over-determined system of equations which is solved by a least squares method to obtain the best estimates of GF ,  al and bl . It should be noted that the trilinear method proposed by Karihaloo et al. [39] cannot be applied in the present study because the specimens have been tested with two notch to depth ratios only, as required by the bilinear model of Karihaloo et al. [15]. It is however known [40] that the bilinear and trilinear approximations give nearly the same values of the size-independent specific fracture energy ðGF Þ. 3. Experimental programme 3.1. Materials Locally available type II cement (CEM II/B-V 32.5R according to EN197-1:2011) and ground granulated blast furnace slag (ggbs) with a specific gravity of 2.95 and 2.40, respectively were used. The super-plasticizer used was a polycarboxylic etherbased type with specific gravity of 1.07. Crushed limestone coarse aggregate with a maximum size of 20 mm and a specific gravity of 2.80 was used, while the fine aggregate was river sand having a specific gravity of 2.65. Limestone powder as filler with maximum particle size of 125 lm was used (specific gravity 2.40). A part of the river sand was replaced by an equivalent volume of the coarser fraction of limestone filler in the size range 125 lm–2 mm. 3.2. Mix design

Mix designation

f cu , 28 days MPa

f st , 28 days MPa

E, 28 days GPa

GF N/m

al mm

30A 30B 30C 60A 60B 60C 80A 80B 80C

35.4 37.0 37.8 60.5 62.9 65.2 79.8 81.6 83.2

2.95 3.04 3.30 3.40 3.52 3.65 4.60 5.00 5.35

33.6 32.7 32.0 36.7 36.6 34.5 42.3 40.8 41.0

132.8 112.3 90.4 143.2 120.3 109.7 146.9 136.5 130.2

49.7 42.4 33.8 43.6 42.6 42.3 50.7 47.9 45.2

8  b  l > b > W > 1  Wa 6 Wl > GF 2ð1 a Þ W < a >    3 2 a b ¼ Gf lþ l  > W W W > > 4 5 1  a P al G 1  > F a > W W 2ð1W Þ :

Three series of SCC mixes were designed according to the mix design method proposed by Karihaloo and Ghanbari [41], Deeb and Karihaloo [42], and Abo Dhaheer et al. [43,44] having 28-day nominal cube compressive strengths of 30, 60 and 80 MPa with w/cm ratios of 0.63, 0.47 and 0.35 respectively. Deeb et al. [45] have also used the same method to design ultra-high strength SCC mixes based on the ultra-high strength vibrated mix CARDIFRC of Benson and Karihaloo [46].The SCC mixes contained different volume fractions of coarse aggregate and paste to solid ratios. They are designated A, B and C for low, medium and high p/s ratios, respectively. The compositions of all mixes are given in Table 1. In order to ensure that all mixes met the flow and passing ability criteria without segregation (SCC requirements), slump flow, J-ring, L-box and V-funnel tests were conducted (Table 2) according to EFNARC guidelines [47]. 3.3. Specimen preparation and test procedure

ð4Þ



To obtain the values of GF , al and bl of a concrete mix, the Gf of specimens of identical sizes and a range (more than three) of the

From each of the nine mixes (Table 1) 12 beam specimens (Fig. 1), three cubes (100 mm), and three cylinders (100  200 mm) were cast. The specimens were de-moulded after 1 day and cured in water at ambient temperature for 28 days. The cube compressive strength was measured according to BS EN 12390-3 [48]. Six of the beams were notched to a depth of 10 mm (notch to depth ratio a/W = 0.1) with a thin (2 mm) diamond saw while the remaining six were notched to a depth of 60 mm (a/W = 0.6). The split cylinder strength (fst) and the modulus of elasticity (E) were measured on cylinders according to BS EN 12390-6 and BS 1881-121, respectively [49,50].

160

GF , N/m

140

120 Grade 80 100 Grade 60 Grade 30 80 26

27

28

29

30

31

32

33

34

Coarse aggregate volume fraction, % Fig. 5. Variation of GF of SCC mixes of different grades with coarse aggregate volume fraction.

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GF

160

fcu

140

140

120

120

100

100

80

80

60

60

40

40

20

20

0

fcu , MPa

G F , N/m

160

0 30A

30B

30C

60A

60B

60C

80A

80B

80C

Mix designation Fig. 6. Variation of the GF and f cu with different p/s ratios.

160

G F , N/m

140

120

33% CA of mix vol

100

30% CA of mix vol 27% CA of mix vol 80 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

w/cm Fig. 7. Variation in GF with w/cm ratio for different coarse aggregate (CA) volume fractions.

4. Results and discussion

Fig. 8. Bilinear softening diagram.

As schematically shown in Fig. 1, the tests for the determination of the fracture energy were performed according to the RILEM work-of-fracture method [3]. The crack mouth opening displacement (CMOD) was used as the feedback control signal and the load-point deflection was measured simultaneously by means of a linearly variable displacement transducer (LVDT). The tests were performed in a stiff Dartec closed-loop universal testing machine with a maximum load capacity of 250 kN.

Typical recorded load–deflection diagrams of two notched beams (out of twelve) from three of the nine mixes are shown in Fig. 4. The area under the load–deflection diagram was calculated from which the Gf (a, W) was determined using Eq. (1). Table 3 shows the results of the measured fracture energy Gf (a, W), with an indication of the mean value, standard deviation and the coefficient of variation (COV%). The specific size-independent fracture energy (GF Þ and the transition ligament lengths (al ) of all mixes are determined from Gf (0.1) and Gf (0.6) of Table 3 using the first of the two equalities in Eq. (3). In many mixes, however it transpired that the transition ligament length al so calculated violated the corresponding inequality for a/W = 0.6. In these cases the first of the two equalities was used only for a/W = 0.1, while the second equality was used for the deeper notch a/W = 0.6. The resulting values of GF and al are reported in Table 4, together with the cube compressive strength, split cylinder strength and modulus of elasticity, measured according to the relevant British standards. Within the range of coarse aggregate volume fraction (27–33%) investigated in this study, GF increases with the increase of coarse aggregate fraction as is clear from Table 4 and Fig. 5 because of the

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5.0

3.0 2.0 1.0

0.6 0.4

0.0 0

0.2

0.4

0.6

0

0.8

0.1

0.2

0.3

0.4

0.5

CMOD, mm

CMOD, mm

(a) 30A: a/ W = 0.1

(b) 30A: a/ W = 0.6

6.0

1.4

Test Model

5.0

0.6

Test Model

1.2 1.0

4.0

Load, kN

Load, kN

0.8

0.2

0.0

3.0 2.0

0.8 0.6 0.4

1.0

0.2

0.0

0.0 0

0.1

0.2

0.3

0.4

0.5

0

0.6

0.1

0.2

0.3

CMOD, mm

CMOD, mm

(c) 60B: a/ W = 0.1

(d) 60B: a/ W = 0.6

10.0

2.0

Test Model

6.0 4.0

0.4

Test Model

1.6

Load, kN

8.0

Load, kN

Test Model

1.0

Load, kN

4.0

Load, kN

1.2

Test Model

1.2 0.8 0.4

2.0

0.0

0.0 0

0.1

0.2

0.3

0.4

0

0.1

0.2

CMOD, mm

CMOD, mm

(e) 80C: a/ W = 0.1

(f) 80C: a/ W = 0.6

0.3

Fig. 9. Load–CMOD curves generated by the hinge model and average experimental load–CMOD curves.

increase in the energy dissipation mechanisms (micro-cracking, crack branching, aggregate interlock) in much the same manner as in VC [1,51–52]. This observation is in agreement with previous research on SCC [28,30]. Moreover, it can be seen from Fig. 5 that the increase in GF with the coarse aggregate volume fraction is less pronounced in the high strength mix (grade 80) than in mix grades 30 and 60. This is may be attributed to the fact that the ITZ [29] in grade 80 mixes is much denser and therefore more susceptible to cracking because it contains a higher proportion of cementitious materials, as can be seen in Table 1.

An increase in the paste to solids (p/s) ratio in all mix grades, as expected, leads to a slight increase in the cube compressive strength fcu but a noticeable decrease in GF , as shown in Fig. 6. As expected, GF decreases with increasing water to binder (w/cm) ratio, in much the same manner as in VC [52,53] as shown in Fig. 7. This result is consistent with the recent study on normal strength self-compacting concrete conducted by Beygi et al. [31] who found that fracture energy decreases by 38% as w/cm ratio is increased from 0.4 to 0.7. GF of high strength self-compacting concrete (fcu  100 MPa), on the other hand has been reported by Cifuentes and Karihaloo [32] to be just 90 N/m for w/cm = 0.23.

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W.S. Alyhya et al. / Construction and Building Materials 110 (2016) 312–322

1.0 Mix A Mix B

0.8

Mix C 0.6 0.4 0.2 0.0 0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

w, mm

(a) Grade 30 1.0 Mix A Mix B

0.8

Mix C 0.6 0.4 0.2 0.0 0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

w, mm

(b) Grade 60 1.0 Mix A Mix B

0.8

Mix C 0.6 0.4 0.2 0.0 0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

w, mm

(c) Grade 80 Fig. 10. The normalised bilinear stress-crack opening relationship for different SCC grades corresponding to their size-independent fracture energy (GF ).

This is a consequence of the densification of ITZ as a result of using a fairly high volume fraction of micro-silica. 4.1. Bilinear tension softening diagram To complete the determination of the fracture properties of the SCC mixes, we now outline briefly an inverse procedure based on the non-linear hinge concept for identifying the parameters of

the bilinear tension softening diagrams of the mixes corresponding to their above size-independent values of the specific fracture energy ðGF Þ. The details of the procedure may be found in [19,20]. It should be mentioned that the popularity of the bilinear approximation of the tension softening diagram (Fig. 8) stems from the fact that it captures the two major mechanisms responsible for the observed tension softening in a concrete mix, namely microcracking and frictional aggregate interlock. The initial linear branch

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Table 5 Parameters of the bilinear softening diagram corresponding to the size-independent specific fracture energy (GF Þ. Mix designation

a1 mm1

a2 mm1

w1 mm

wc mm

ft

r

GF N/m

E GPa

lch mm

30A 30B 30C 60A 60B 60C 80A 80B 80C

10.07 13.21 18.76 10.97 13.17 15.38 16.19 18.51 19.75

1.12 1.05 1.20 1.14 1.25 1.28 1.18 1.39 1.38

0.078 0.060 0.043 0.073 0.062 0.053 0.048 0.043 0.041

0.272 0.254 0.198 0.251 0.213 0.198 0.238 0.194 0.177

0.218 0.203 0.186 0.203 0.189 0.185 0.225 0.211 0.188

132.8 112.3 90.4 143.2 120.3 109.7 146.9 136.5 130.2

33.6 32.7 32.0 36.7 36.6 34.5 42.3 40.8 41.0

1377 976 519 1057 771 497 647 380 289

Table 6 Relation between f ct and f st of test SCC mixes. Mix designation

f ct MPa

f st MPa

f ct f st

Mean

30A 30B 30C

1.80 1.94 2.36

2.95 3.04 3.30

0.61 0.64 0.72

0.66

60A 60B 60C

2.23 2.39 2.76

3.40 3.52 3.65

0.66 0.68 0.75

0.69

80A 80B 80C

3.10 3.83 4.30

4.60 5.00 5.35

0.67 0.77 0.80

0.75

f ct f st

of the bilinear diagram which is steep is a consequence of the micro-cracking, whereas the second linear branch which is shallow is a result of the frictional aggregate interlock. In the non-linear hinge model of a pre-notched beam, a part of the beam on either side of the notch is isolated as a short beam segment subjected to a bending moment and a normal force. The growth of the real crack and associated fictitious crack representing the fracture process zone is viewed as a local change in the overall stress and strain fields in this isolated beam segment. The constitutive relationship inside the hinge segment depends on the position of the fictitious crack along the depth of the beam. The axial load and bending moment are related to the hinge rotation in four phases depending on the crack propagation. Phase 0 represents the elastic state when no fictitious crack has formed ahead of the pre-existing notch. Phases I, II and III represent different stages of crack propagation. In phase I, the fictitious crack of length d ahead of the notch is such that the maximum crack opening is less than w1 corresponding to the knee in the bilinear

diagram. In phase II, a part of the fictitious crack of length longer than d has a crack opening in excess of w1, but in the remaining part it is less than w1. In phase III, a part of the fictitious crack has opened more than wc and thus become traction-free, while the opening of the remaining part is still less than wc or even w1. Analytical expressions relating the hinge rotation to the bending moment and crack length in each phase, and in turn to the applied central load on the beam and crack mouth opening displacement (CMOD) are given in Abdalla and Karihaloo [19]. The expressions for CMOD and central load are used to minimise the sum of squares of the errors between the theoretical and experimental values of the load with respect to the three unknown parameters of the bilinear tension softening diagram (Fig. 8). The accuracy of this minimization procedure depends on the total number of observations from the recorded load–CMOD diagram used in this procedure and the allowable error (<3%). Typical results of this minimization procedure are shown in Fig. 9 for some SCC mixes and pre-existing notch depths. As the load–CMOD diagrams are recorded on tests on beams with a notch to depth ratio of 0.1 or 0.6, the three unknown parameters of the bilinear tension softening diagram obtained from the above minimization procedure correspond not to the GF of the SCC mix but to its size-dependent Gf (0.1) and Gf (0.6). These pairs of three parameters need therefore to be appropriately scaled to reflect the size-independent GF of the mix. The scaling procedure is described in Abdalla and Karihaloo [19]. The bilinear tension softening diagrams of all nine SCC mixes corresponding to their size-independent specific fracture energy ðGF Þ are shown in Fig. 10 and tabulated in Table 5. The three parameters describing the shape of the bilinear diagram, together with the direct tensile strength (fct) and the elastic modulus (E) of all SCC mixes are given in Table 5. The slope of the initial part of the bilinear softening

0.85 Mix A

Mix B

Mix C

0.80

fct /fst

0.75 0.70 0.65 0.60 0.55 0.50 Grade 30

Grade 60

Grade 80

Fig. 11. Direct (fct) and indirect (fst) tensile strengths of different SCC mixes.

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1600 Mix A

Mix B

Mix C

1400 1200 1000 800 600 400 200 0 Grade 30

Grade 60

Grade 80

Fig. 12. Characteristic length (lch) of different test SCC mixes.

curve increases with the increasing the p/s ratio, but the influence of p/s decreases as the f cu of the mix increases. It is well documented that the direct tensile strength (f ct ) is approximately two thirds of the indirect tensile strength (f st ). Although the literature is rich in reporting on SCC, the effect of p/s ratio and mix grade on tensile strength is still not fully addressed. The relationship between the direct tensile strength (determined by the inverse analysis using the non-linear hinge model) and splitting strengths (f ct /f st ) of SCC mixes of different p/s ratio and mix grade are summarised in Table 6 and Fig. 11. It is found that f ct /f st is dominated by the p/s in the mix and the mix grade: it increases with both an increase in p/s and mix grade. This might provide a better understanding of the contribution of p/s and strength on the tensile strength of SCC and a useful guide for determining the f ct from the f st in SCC mixes. Note that the ratio (f ct /f st ) is slightly different from the conventional 0.65 [54]. It depends on the p/s ratio and strength grade (Table 6). Also given in Table 5 is the characteristic length (lch) of each mix calculated using the relation:

lch ¼

EGF 2

f ct

ð5Þ

The characteristic length represents the ductility of a mix; the larger the characteristic length, the more ductile the mix. lch is dominated by the coarse aggregate volume fraction and it decreases with increasing strength grade (Fig. 12). 5. Conclusion Based on the results of this study, the following conclusions can be drawn: 1. The results confirm the dependency of the RILEM fracture energy on the notch depth. 2. The specific fracture energy (GF ) increases with an increase in the coarse aggregate volume fraction, irrespective of the SCC mix grade, although the increase is less pronounced in higher strength mix (grade 80) than in grades 30 and 60 of SCC. 3. Within the same nominal strength grade, an increase in the paste to solids (p/s) ratio results in a marginal increase in the strength itself, but a noticeable decrease in GF . 4. An increase in the w/cm ratio reduces GF . The decrease becomes more pronounced with decreasing coarse aggregate volume fraction.

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