Analysis of material size and shape effects for steel fiber reinforcement self-consolidating concrete

Engineering Fracture Mechanics 206 (2019) 46–63

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Analysis of material size and shape effects for steel fiber reinforcement self-consolidating concrete M. Akbaria, S. Khalilpourb, M. Dehestanic, a b c

T

⁎

Mazandaran University of Science and Technology, Babol, Iran Babol Noshirvani University of Technology, Babol, Iran Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran

A R T IC LE I N F O

ABS TRA CT

Keywords: Fracture mechanics Size effect law: SCC LEFM Compressive strength

An experimental program is conducted in this study to evaluate the specimen shape and size effects on the compressive strength of steel fiber reinforced self-consolidating concrete (SFRSCC). The results of fresh and hardened concrete properties are first examined through casting cubic specimens in five different sizes and three sizes of cylindrical specimens with a height-to-diameter ratio of 2. The shape-effect model is then evaluated and the coefficients are obtained. Furthermore, relations between cubic and cylindrical specimens are presented and suggestions for the practical cases are offered. Subsequently, the size effect models SEL, MSEL, MFSL, USEL, and the model proposed by Sim et al. are discussed and the coefficients for self-consolidating concrete (SCC) and SFRSCC are updated. Moreover, cubic and cylindrical specimens are compared regarding crack pattern. The results clearly express that with the increase of steel fibers, the size effect is decreased and the structure becomes more ductile. The size effect in cubes is more pronounced than that of the cylinder. By adding and increasing the steel fiber content, the size effect decreased owing to crack bridging and the reduction of crack growth. Also, failure moves towards the strength criteria region in the SEL and towards the Euclidian regime in the MFSL.

1. Introduction Microcracks are generated in the concrete structures owing to some conditions and defects during construction, maintenance, and loading. These cracks begin to join after loading and then propagate once reaching their critical condition level. Therefore, fracture mechanics is employed to better understand the process of crack formation and propagation. Yet, what matters is the difference in the behavior of concrete structural members compared to laboratory specimens due to the size effect phenomenon. In addition, change in the cross-sectional geometry of structural members relative to laboratory specimens is of great importance. This is because of the difference in the failure mode, thus addressing the specimen shape effect. The effect of shape and size on the compressive strength of conventional and high strength concrete was primarily examined and equations were presented for converting them to each other [1–3]. Neville [4,5], studied the effect of shape and size and proposed a relation to convert cylindrical specimens to cubic ones. Despite all studies on the size and shape effects based on the geometry of cross-section, Bazant 1984 [6] provided a relation to predict the size effect for specimens of similar shapes and different sizes, using the energy equilibrium in crack extension on the basis of fracture mechanics termed as the size effect law (SEL). In recent years, numerous studies have been conducted on beams in order to

⁎

Corresponding author at: Postal Box: 484, Babol 47148-71167, Iran. E-mail address: [email protected] (M. Dehestani).

https://doi.org/10.1016/j.engfracmech.2018.11.051 Received 16 July 2018; Received in revised form 24 November 2018; Accepted 26 November 2018 Available online 27 November 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.

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examine the size effect in structural members. This led to an enhancement in the prediction of the failure behavior in different types of concrete, the understanding of the element behavior in real dimensions, the crack propagation process and the amount of concrete brittleness [7–12]. CEP-Fip 1990 [13], declared that with increasing compressive strength from 40 to 80 MPa, size effect decreases and the strength ratio is approximately halved. In 1990, Kim et al. [14] investigated the size effect of concrete structures without initial cracks. They believed that initial cracking in large members would lead to a significant reduction in compressive strength, while this reduction would not be severe in the case of no initial cracks. Kim and Eo 1990 [15], corrected the size effect law in the same year, which was appropriate to estimate the size effect of compressive, tensile and shear strengths. They used nonlinear fracture mechanics and the regression analysis on an extensive range of specimens to propose their relation. Jansen and Shah 1997 [16], investigated the size effect in cylindrical specimens with a height-to-diameter ratio (h/D) changing from 2.5 to 5.5 and specified that with increasing (h/D), the behavior of stress-strain curve varies and the post-peak slope increases. Carpinteri and Chiaia 1997 [17], offered a relation named as the multifractal scaling law (MFSL). They employed the fractal theory to determine the size effect. Tokyay and Özdemir 1997 [18], investigated the effect of size on the specimens with strength ranging from 40 to 75 MPa and concluded that the size effect has decreased with increasing strength. Kim et al., 1990–2005 [19–22], examined the modified size effect law (MSEL) in cracked and uncracked cylindrical specimens subjected to uniaxial compressive force, axially loaded double cantilever beam (DCB), C-shaped specimens subjected to flexural compression force, flexural compressive strength of reinforced concrete beam and compressive strength of spirally reinforced concrete cylinders. Specimen size effects on the compressive strength of concrete at the bearing area were studied by Ince and Arici, 2004 [23]. Yi et al. 2006 [24], reviewed the compressive strength size and shape effects of cylindrical, cubic and prism specimens subjected to normal and parallel placement, and reported that the size effect in cube and prism is more than that of the cylinder. Viso et al. 2008 [25], figured the effects of size and shape on cubic and cylindrical specimens at maximum load (SEL), the stressstrain curve and the crack pattern in specimens. The results revealed that the size effect was stronger in cube compared to the cylinder. Similarly, in the study of the strain-stress curve in specimens smaller than the standard size, the maximum strain at peak load has increased (reaching approximately 0.005). Thus, the failure pattern in the cube was spalling and hour-glass shape, while being conic in the cylinder. Accordingly, the failure in the cube was denser than in cylinder. Sim et al. 2013 [26], suggested a relation to predict the size effect based on the fracture mechanism by carrying out a comprehensive study of lightweight concrete. Nikbin et al. 2014 [27], investigated the effect of placing on the size effect in self-consolidating concrete (SCC) and stated that the cube-tocylinder ratio in SCC is higher than that of conventional concrete. In recent years, different models based on the geometry of section, the fracture mechanics (SEL, MSEL, Sim et al. [26]) and the fractal theory (MFSL) have evaluated the size effect in different types of concrete and have improved the size effect coefficients for many concrete types [28–31]. The most important reason for the use of fibers is to improve the mechanical performance of concrete in order to increase its energy absorption capacity [32–34]. The addition of fiber improves the durability and mechanical properties of hardened concrete such as flexural strength, brittleness, impact strength, resistance to fatigue, and vulnerability to cracking and spalling [35–37]. Fibers generally bridge the microcracks and hamper the crack propagation. When the tensile stress is transferred to fibers, the transfer can arrest the propagating macrocracks and substantially improve the tensile strength of concrete [38–40]. Steel fibers are also used in concrete in order to increase the energy absorption capacity and to create a ductile concrete structure. 1.1. Research significance One of the advantages of SCC is its high workability which can fill spaces between the reinforcing bars and then minimize the problem of vibration in the load-carrying members. When steel fibers are added to the concrete, its workability is reduced, however, the mechanical properties improve. Consequently, a limited volume percentage is used to maintain the workability of SCC. On the other hand, one of the main sources of size effect in concrete structures is the presence of fracture process zone (FPZ), in which a large zone of microcracking in concrete blunts the fracture front. This causes size effect in concrete [41,42]. Therefore, one can expect that the steel fibers, as one of the best fiber types for bridging cracks and thus creating a coherent zone to prevent crack propagation, reduce size effect. Also, due to the different characteristics of steel fiber reinforced self-consolidating concrete (SFRSCC), the size effect varies depending on the properties of materials including compressive strength, tensile strength, shear strength, etc. The compressive strength size effect is quantitatively investigated in the present study. 2. Materials and methods The ACI 237R-07 [43] was adopted for mix designs in this study. Portland cement of type II was used in all mixtures. Fine aggregate was sand with a fineness modulus of 2.73, a specific density of 2.61 g/cm3, a water absorption of 1.94% and a maximum size of 2.36 mm. In addition, crushed coarse aggregate with a maximum diameter of 12.5 mm and a specific density of 2.68 g/cm3 and 1% water absorption was used. The polycarboxylate superplasticizer was also used to improve the flowability of concrete. For dispersion, limestone powder filler was added with a specific gravity of 2666 kg/m3 and a specific surface area of 480 m2/kg. To improve the mechanical properties of concrete specimens, steel fiber with a length of 33 mm, and ultimate tensile strength of 1220 MPa were used with volume fractions of 0.1, 0.2 and 0.3 as shown in the Table 1. The designs were mixed according to ACI 237R-07 [43] and then placed in the test specimens. The specimens were removed from the mold and transferred to the water pool for storage after 1 day. Slump flow, T500, L-box, and V-funnel tests were performed on SCC in order to respectively measure the concrete flowability (size of 2 perpendicular diameters), the deformation rate via defining a flowability distance, segregation and viscosity of concrete, and finally to simulate the flow rate of concrete between the reinforcing bars on each of the four plain and the fibrous 47

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Table 1 Mix proportions. Mix design

Water (kg)

Cement (kg)

W/C

Coarse agg (kg)

Fine Agg (kg)

Limestone Powder (kg)

Steel Fiber (%)

Superplasticizer (kg)

SCC SCC_0.1SF SCC_0.2SF SCC_0.3SF

202.5 202.5 202.5 202.5

450 450 450 450

0.45 0.45 0.45 0.45

700 700 700 700

836 836 836 836

240 240 240 240

0 0.1 0.2 0.3

1.8 1.8 1.8 1.8

designs [44]. Fig. 1a shown that the Cubic and cylindrical specimens (h/d = 2) were constructed in different dimensions to examine the shape and size effect models and to improve their relations. Details of the specimens are presented in Table 2. In order to investigate the compressive strength size and shape effects, pressure tests were carried out subjected to a 200-KN pressure jack that can be seen in Fig. 1b. Cubes were loaded perpendicular to the placement direction, according to BS-EN 12,390 [45] and the cylinders were loaded parallel to the placement direction, conforming to ASTM C39 [46].

2.1. Experimental results Table 3 shows the results of experiments on fresh SCC concrete. The results revealed that by adding steel fibers, the rheological qualifications of concrete decrease significantly. Adding 0.1, 0.2 and 0.3% of the fiber to concrete, respectively decreased the slump flow by 0%, 5.71% and 8.82%, increased slump time by 4.76%, 26.98% and 50.79%, increased the V-Funnel time by 27.2%, 49.71% and 74.34%, and decreased the L-Box results by 3.16%, 8.89%, and 18.07%. However, all results are still in their standard range. The specimens were removed after 28 days of curing in the water pool. Table 4 presents the results of compressive strength tests, in which the strength of cubes with same cross-sectional dimension, is greater than that of cylinders. Fig. 2 illustrates the trend of variation of compressive strength by changing the diameter of cylinders and cubes. As expected, the compressive strength decreased with the increase of diameter. Furthermore, with an increase in the fraction of fiber volume, compressive strength slightly increased, as shown in Fig. 3. Indeed, the results indicate that the microcracks are bridged via steel fibers, leading to a reduction in the crack propagation process, and thus preventing cylinders from conical and diagonal failures, and the cubes from spalling failure as signified in Fig. 4. Furthermore, in the case of cubes, the fracture process is provoked by a stress concentration near the cube corners and micro-cracks connecting together in this place, which appeared with a lot of cracking, leading to the hourglass. This result shows that cubic specimens had denser and more micro-cracks compared with cylindrical specimens.. Moreover, the number of microcracks at the fracture surface of specimens containing steel fiber was greater than that of non-fibrous specimens.

Fig. 1a. Cubic and cylindrical specimens in various sizes. 48

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Table 2 Specification of specimens. Shape

Dimension (mm)

Cylinder (cy) Cube (cu)

∅75 × 150, ∅100 × 200, ∅150 × 300 (50 × 50 × 50), (75 × 75 × 75), (100 × 100 × 100),

(125 × 125 × 125),

(150 × 150 × 150)

Fig. 1b. A representative of the failure of specimens using load-control devise. Table 3 Fresh properties of self-consolidating concrete mixtures. Mix

V-Funnel (sec)

SCC SCC_0.1SF SCC_0.2SF SCC_0.3SF

13.64 17.35 20.42 23.78

Slump Flow

L-Box (h2/21)

T500 (sec)

D-Final (mm)

3.15 3.3 4 4.75

740 740 700 680

0.98 0.95 0.9 0.83

3. Models of size and shape effects 3.1. Models based on specimen geometry Neville 1966 [5], presented a relation for converting cubes to cylinders and prisms. He used cylinders with (h/d) = 1, 2; prisms with (h/d) = 2, 3; and results of 11 existing experimental programs for cubes. However, this relation was based on the geometry of the sections and thereby disregarded the crack propagation. Correspondingly in the previous years, regulations and standards of different countries considered coefficients for the conversion of specimens including ASTM [46], AASHTO [47], United States Bureau of Reclamation, and UNESCO [48]. Neville presented his relation as follows:

P = P6

α V 6hd

+

h d

+b (1)

where P is the compressive strength of cylindrical specimen, P6 is the strength of 6 in (150 mm) cube specimen, V is the volume of the specimen (in3), d is the maximum lateral dimension (in) and h/d is the height-to-diameter ratio. Dehestani et al. 2014 [29], employed P6 for cylinders and established a relation between different cylindrical specimens 49

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Table 4 Results of compressive strength (a): cubes (b): cylinders. (a): Cubes Mixture code Size (mm)

Compressive strength (MPa)

Mix design

Size (mm)

Test 1

Test 2

Test 3

Ave

SCC

50 × 50 × 50 75 × 75 × 75 100 × 100 × 100 125 × 125 × 125 150 × 150 × 150

51.8 47.3 46.7 42.5 40.1

52.4 48.1 45.9 43.7 39.3

52.7 49.7 43.4 39.4 38.6

52.3 48.3 45.3 41.8 39.3

SCC_0.1SF

50 × 50 × 50 75 × 75 × 75 100 × 100 × 100 125 × 125 × 125 150 × 150 × 150

56 52.4 48.6 45.9 43.0

53.3 50.9 46.4 43.5 39.4

54.8 50.3 49.4 44.2 42.0

54.7 51.2 48.1 44.5 41.6

SCC_0.2SF

50 × 50 × 50 75 × 75 × 75 100 × 100 × 100 125 × 125 × 125 150 × 150 × 150

60.4 53.7 51.3 48.5 45.6

58.0 51.4 49.6 48.7 44.3

56.7 54.0 50.8 46.7 44.1

58.3 53.0 50.5 47.9 44.6

SCC_0.3SF

50 × 50 × 50 75 × 75 × 75 100 × 100 × 100 125 × 125 × 125 150 × 150 × 150

61.4 56.5 50.4 49.5 47.3

60.1 58 53.7 49.1 48.2

62.7 54.2 51.6 48.5 46.8

61.4 56.2 51.9 49.0 47.4

(b): Cylinders Mixture code Size (mm)

Compressive strength (MPa)

Mix design

Diameter (mm)

Height (mm)

Test 1

Test 2

Test 3

Ave

SCC

75 100 150

150 200 300

43.7 40.3 37.4

41.8 38.1 34.5

41.3 39.7 36.6

42.2 39.3 36.1

SCC_0.1SF

75 100 150

150 200 300

45.4 43.3 41.1

43.5 41.0 40.3

45.1 43.9 38.9

44.6 42.7 40.1

SCC_0.2SF

75 100 150

150 200 300

46.7 45.0 42.2

45.3 44.7 42.6

45.9 43.0 39.4

45.9 44.2 41.4

SCC_0.3SF

75 100 150

150 200 300

48.0 45.1 42.1

45.9 46.4 43.0

46.8 44.2 41.9

46.9 45.23 42.23

compared to the standard one for SCC concrete with 2 different water-to-cement ratios. The least squares method (LSM) is employed in this study to obtain the coefficients in the proposed relations. This method can be used to obtain relations with at least 2 unknown parameters, while the best fit delivers coefficients as outputs. Table 5 summarizes the coefficients of relation (1) corresponding to SFRSCC. The results display no significant change in R2 after the addition of steel fibers. Despite the fact that both shapes of specimens have been broken up as expected, that is, cylindrical specimens broke as column failure or in the conic form, and cubic specimens broke in the form of hourglasses. Since the micro-cracks are the most important reason of size effect, steel fibers can however reduce the size effect by connecting micro-cracks and decrease size effect while no reduction in shape effect occurs. Fig. 5 exhibits the obtained experimental value and the predicted one, indicating that the shape effect is not reduced with the addition of fibers, thus having a relatively large scatter. Besides, R2 values were about 0.60. As a result, it can be inferred that the above relation is not suitable for predicting shape effect in the SCC as well as the fiberreinforced self-consolidating concrete (FRSCC). Kim and Yi 2002 [49], proposed a relation for determining the size effect in cylinders of different diameters relative to the standard, based on the nonlinear fracture mechanics. Similar to Neville's 1966 [5] relation, it did not account for the effect of aggregate, specimen weight and other influential factors, addressing only the effect of (h/d). They presented their relation as follows:

f0 f c'

=

α 1+

(h − d) 50

+b (2) 50

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65 60

fc (MPa)

55 50 SCC

45

SCC-0.1SF

40

SCC-0.2SF SCC-0.3SF

35 40

60

80

100

120

140

160

Dimension (mm)

(a) 47

fc (MPa)

44 41 SCC

38

SCC-0.1SF SCC-0.2SF

35

SCC-0.3SF

32 60

70

80

90

100

110

120

130

140

150

160

Diameter (D) (mm)

(b) Fig. 2. Variation of relative compressive strength with different specimen sizes: (a) Cube specimen; (b) cylinder specimen.

where f0 is the compressive strength of the cylindrical specimen with different dimensions (MPa), f'c is the compressive strength of the standard cylindrical specimen (MPa), h is the height of specimen (mm) and d is the diameter of the specimen (mm). Fig. 6 depicts that the predicted and experimental values in the concrete under consideration are in reasonably good agreement and therefore suitable for estimating the size effect. Due to the high flowability of SCC, its pores are reduced and materials are best placed together, causing the concrete to fail once the failure stress is reached. Given that in size effect model of small specimens, the failure pattern is very determinative, so that the use of SCC results in the same failure patterns and hence the size effect can be readily estimated. Although the fibers reduce crack growth, the normal SCC model yields a similar result in cylindrical specimens as well. The obtained constants and R2 values presented in Table 5 give rise to this point. Fig. 7 shows the relation between the size and shape effects normalized for three same cross sections (75, 100, 150 mm) using the LSM, the best function between points is visible. Polynomial regression equations can be used in practical projects or research where a designer or researcher intends to use these four designs. The designer can estimate the strength of cylindrical specimen using the strength of cube specimen. Relations have been presented for cubic and cylindrical specimens in numerous works [18,25,29,50,51]. In this paper, the relation between cube and cylinder with the same geometry is presented. Fig. 8 highlights the variations in the compressive strength of cylindrical specimens versus cubic specimens of dimensions 75, 100 and 150 mm. In order to determine the highest correlation between dimensions, all information on the four designs with the cited dimensions were compared and then a linear fit was passed through. A linear fit is a straight line that is passed by using the LSM among the given × and y data indicating the tendency of the points. The relations showed that the 100-mm specimen with R2 = 0.80 has the best performance among the four designs and then the 150-mm specimens with R2 = 0.69 and the 75-mm specimens with R2 = 0.61, respectively. In other words, the 100-mm specimen has the maximum correspondence of compressive strength in SCC and SFRSCC designs. Therefore, it is recommended that when using SCC and FRSCC in laboratory tests for the structural applications, a 100-mm cylinder is used for elements with cubic cross sections including columns. Some standards, regulations, and articles have also put forward recommendations. British standard BS 1881-116 [52] suggests 100 and 150-mm cubic specimens. Eurocode [53] offers both cubes and cylinders, while ASTM C39 [46] and ASTM C31 [54] propose cylindrical specimens of 150 × 300 and 200 × 100, respectively. Alternatively, regulations of some countries like 51

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65

fc (MPa)

60 55 50

50mm 75mm

45

100mm

40

125mm 150mm

35 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Steel Fiber (%)

(a) 48 46

fc (MPa)

44 42 40

75mm

38

100mm

36

150mm

34 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Steel Fiber (%)

(b) Fig. 3. Variation of relative compressive strength with different steel fiber (%): (a) Cube specimen; (b) cylinder specimen (diameter).

Fig. 4. Modes of failure of specimens of different sizes (a): cubes (b)“ cylinders.

Germany [24] suggest 150-mm cubes, Norway [24] proposes 150-mm cubes and 150 × 300 cylinders, while the Australian standard [55] only uses cylindrical specimens of 150 × 300 and 200 × 100. 3.2. Models based on fracture mechanics In the 1920s [56], Griffith proposed the LEFM by ignoring the crack tip plasticity. This approach could be used for large specimen sizes being linear up to failure limit. In the following years, Irwin [57], proposed the concept of the energy release rate and 52

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Table 5 Correlations between strengths of various specimens. (a): 150 mm cube and different cylinder diameter (b): standard cylinder and different cylinder diameter. Model P P6

f0 f c'

=

α V h + 6hd d

+b

α

= 1+

(h − d) 50

(a)

+b

(b)

α

b

R2

1.34621 0.95609 0.89506 0.84319

0.46277 0.64069 0.62693 0.60966

0.60 0.59 0.59 0.60

1.26730 0.85906 0.83611 0.82102

0.36543 0.57023 0.58304 0.59344

0.99 0.99 0.99 0.99

Estimated fcy/fcu(150mm)

1.1

1.05

1

0.95

SCC SCC-0.1SF

0.9

SCC-0.2SF SCC-0.3SF

0.85 0.85

0.9

0.95

1

1.05

1.1

Measured fcy/fcu(150mm) Fig. 5. Comparisons of measured and predicted compressive strength of Neville 1966 [5]

1.18

Estimated f0/f'c

1.13

1.08 SCC SCC-0.1SF 1.03

SCC-0.2SF SCC-0.3SF

0.98 0.98

1.03

1.08

1.13

1.18

Measured f0/f'c Fig. 6. Comparisons of measured and predicted compressive strength of Kim and Yi 2002 [49]

generalized Griffith's theory. Then, Hillerborg et al. 1976 [58], introduced the Fictitious crack model through which the crack behavior could be examined. In the early 1980s, Peterson [59] and Hillerborg [42], employed the fracture energy in lieu of the strain energy release rate, which was based on the energy absorption and cracks formation on the same plane. In this regard, due to many weaknesses of linear elastic fracture mechanics (LEFM), nonlinear fracture mechanics was proposed widely covering the deficiency of the previous method. Yet in this period, all the experiments on the formation and expansion of crack considerably depended on the size of specimen, when Bazant 1984 [6] presented the size effect law (SEL) for the first time on the basis of fracture mechanics, where 53

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1.18 1.16 1.14

fcu/f'c

1.12 1.10 y = 1.7381x2 - 3.54x + 2.7938 R² = 1

1.08

SCC

1.06

SCC1_0.1SF y = 0.8893x2 - 1.5893x + 1.6917

1.04

SCC1_0.2SF

1.02

SCC1_0.3SF

1.00 0.98 1.00

1.05

1.10

1.15

1.20

1.25

1.30

R² = 1 y = 1.0124x2 - 1.8511x + 1.8192 R² = 1 y = -1.3355x2 + 3.807x - 1.5905 R² = 1

1.35

1.40

fcy/f'c Fig. 7. Polynomial regression relation between normalized size and shape effects.

Cube (75mm)

58.00

y = 1.2538x - 4.1521 R² = 0.6149

55.00 52.00 49.00

Data

46.00

y=x

43.00

Linear (Data)

AVG

40.00 40

45

50

55

60

Cylinder (75mm)

(a) 56.00 y = 1.024x + 5.0628 R² = 0.803

Cube (100mm)

53.00 50.00 47.00

Data

44.00

y=x

41.00

AVG

38.00

Linear (Data)

35.00 35

40

45

50

55

Cylinder (100mm)

(b)

Cube (150mm)

48.0

y = 1.0387x + 1.6762 R² = 0.6941

45.0 Data

42.0

y=x

39.0

AVG

36.0

Linear (Data)

33.0 33

36

39

42

45

48

Cylinder (150mm)

(c) Fig. 8. relation between cubes and cylinders (diameter) of different geometry (a):75 mm (b):100 m (c): 150 mm.

54

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the shapes of specimens were the same, however the sizes were different. He assumed that the potential energy release in the fracture is a function of the length and area of the crack band. In the SEL, the difference between the smallest and largest specimens should be at least 4 times and the load-displacement curve is inspected only up to the peak load moment. In fact, the shape change of the curve does not affect the gain of size effect. He presented his relation as follows:

Bfcs

f c (D) =

1+

D D0

(3)

where f c (D) is the compressive strength of cylindrical specimens of different diameters, fcs is the compressive strength of a standard cylindrical and cubic specimens, D is the cylinder diameter, and D0 and B are the empirical constants obtained by the LSM. Kim and Eo 1990 [15], modified the SEL relation by adding the parameter σ0 = αft' which was independent of size, though capable of predicting the size effect of specimens with non-similar initial cracks, as well as an extensive range of uniaxial and splitting tests with different sizes and strengths of very large specimens. They presented their relation based on the initiation and propagation of cracks in the specimen as follows:

Bf 'c

f c (D) =

1+

D λ 0 da

+ Cfcs (4)

where C is a constant, da is the maximum aggregate size taken to be 12.5 mm in this paper, and λ 0 is a constant varying between 2 and 3, assumed to be 2 in the present article. Carpinteri and Chiaia 1997 [17], provided the size effect relation according to the fractal theory that divides the material into two sections of the Fractal regime and the Euclidian regime. One of the conditions for examining size effect in MFSL is to use specimens with h/d = 2. The fractal theory determines the size effect taking material irregularities into consideration which changes with the specimen size, where the microstructural behavior is of minor significance in macroscopic scales. They presented their relation based on certainty and able of predicting large sizes as follows:

f c (D) = f cs∞ 1 +

L ch D

(5)

where f cs∞ is the standard specimen compressive strength with infinite size and L ch is the internal characteristic length. Bazant 1997 [60], employed the concept of fractal theory on size effect law and updated the new model namely, fractal fracture size effect law (FFSEL). It was believed that within a certain range of sizes, the fracture surfaces in a number of materials such as rock, concrete, and ceramics show kinds of fractal characteristics. He presented his relation as follows:

f c (D) =

σ0 D (

df − 1 ) 2

1+

d D0

(6)

where σ0 is strength for a sample with negligible size, which may be expressed in terms of an intrinsic strength, df fractal dimension relies on the fractal characteristics that has positive values df = 1 for nonfractal characteristics, and df ≠ 1 for fractal characteristics. After about 17 years, Masoumi et al. in 2015 [61], worked on SEL and FFSEL, and improved size effect law based on fractal theory and introduced a unified size effect law (USEL). For this end, they investigated the laboratory tests which had been performed on Gosford sandstone samples having a range of sizes of 19, 25, 31, 50, 65, and 96 mm as well as the other researchers. They believed that the size effect in the small sample has the inverse result, which means a peak point can be seen in the curve of size effect that has maximum uniaxial compressive strength. In other words, they divided the size effect diagram into two parts of descending and ascending sections. Then, they presented a relation derived from SEL and FFSEL which reads:

Bf di = ⎛ cs ⎞ ⎝ σ0 ⎠ ⎜

2 df − 1

⎟

(7)

where di is the intersection point of descending and ascending sections in which maximum strength is observed. Two years later, they stated that MFSL is more useful than SEL, since it can consider a broad range of specimen sizes compared to SEL. They then proposed an improvement to unified size effect law for intact rock (IUSEL) [62]. At first, they modified MSFL as fallows:

f c (D) = fmD

(df − 1) 2

1+

L ch D

(8)

where fm is a characteristic strength. Then, they presented their relation that writes:

f di = ⎛ cs∞ ⎞ ⎝ fm ⎠ ⎜

2 df − 1

⎟

(9)

Furthermore, the maximum strength at di is defined as: 55

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Table 6 Results of regression analyses. (a): cubes (b):cylinders. (a): Cubes Model

Mix design

Coefficients A

B

SEL

SCC SCC_0.1SF SCC_0.2SF SCC_0.3SF

1.837 1. 828 1. 637 1. 619

MSEL

SCC SCC_0.1SF SCC_0.2SF SCC_0.3SF

1.64 1.57 1.46 1.51

MFSL

SCC SCC_SF 0.1 SCC_SF 0.2 SCC_SF 0.3

Sim et al. 2013

SCC SCC_0.1SF SCC_0.2SF SCC_0.3SF

C

X2

X4

f∞

61.95 67.59 87.58 95.61 0.39 0.42 0.46 0.42

0.92 0.90 0.91 0.92

−0.33 −6.81 0.12 0.43

0.12 0.02 0.10 0.21

R2 0.91 0.90 0.92 0.95

32.94 35.59 38.20 39.03 1.25 2.37 0.99 1.02

Lch

35.40 58.52 34.79 24.79

79.60 72.06 68.20 75.21

1.75 3.60 1.19 1.15

0.99 0.99 0.99 0.99 0.90 0.92 0.98 0.84

(b): Cylinders Model

Mix design

Coefficients A

B

SEL

SCC SCC_SF 0.1 SCC_SF 0.2 SCC_SF 0.3

1.59 1.51 1.38 1.31

MSEL

SCC SCC_SF 0.1 SCC_SF 0.2 SCC_SF 0.3

1.37 0.93 0.90 0.89

MFSL

SCC SCC_SF 0.1 SCC_SF 0.2 SCC_SF 0.3

Sim et al. 2013

SCC SCC_SF 0.1 SCC_SF 0.2 SCC_SF 0.3

strength = f cs∞ 1 +

C

X4

f∞

91.12 110.65 150.63 209.26

2.49 0.15 0.02 0.16

R2 0.83 0.77 0.81 0.89

0.47 0.64 0.65 0.66

0.82 0.76 0.82 0.84 28.84 35.05 36.46 37.43

4.39 0.77 1.58 0.77

Lch

−160.06 29.65 53.98 26.98

0.59 0.62 −2.36 0.63

14.42 0.29 2.39 0.29

86.16 47.24 45.01 43.62

0.99 0.99 0.99 0.99 0.82 0.77 0.80 0.83

L ch

( )

2

f cs∞ df − 1 fm

(10)

Sim et al. 2013 [26], proposed a relation by examining a wide range of specimens in different dimensions, in order to predict the size effect on the basis of the energy balance during the propagation of the crack band, which forms from smeared microcracks rather than from a single-line crack. Some advantages of this method are the ability to predict the effect of size with h/d = 1, 2 and also to express the distribution of cracks in a specimen compared to other ones. They provided their relation for the size effect used in uniaxial pressure as follows:

f c (D) =

A (n) x 4 f 'c ρ

1 + BD( ρc )−x2

f 'c + Cfcs (11)

0

in which A, x2, and x4 are constants, n is the height-to-lateral dimension ratio assumed to be 2 for cylinder and 1 for the cube in this paper, ρc is the density of the tested concrete and ρ0 = 2300 kg/m3.

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0.2

strength criterion

Log (fc(D)/Bfcs)

0.0 -0.2 -0.4

SEL (Cu) SCC SCC_0.1SF SCC_0.2SF SCC_0.3SF

-0.6 -0.8 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Log (D/D0)

(a) 0.2

strength criterion

Log (fc(D)/Bfcs)

0.0 -0.2 -0.4 -0.6 -0.8 -1.5

SEL (Cy) SCC SCC_0.1SF SCC_0.2SF SCC_0.3SF -1.0

-0.5

0.0

0.5

1.0

1.5

Log (D/D0)

(b) Fig. 9. Bazant's size effect law model (SEL) for (a):cubes (b):cylinders.

4. Results and discussions Table 6 presents the results of SEL predictions along with experimental ones through LSM. Fig. 9 compares the SEL results indicating that with increasing size, the specimens tend to fail in LEFM mode characterized by an inclined line of slope (-1/2). In addition, as the dimensions of specimens are reduced, specimens shift towards the strength criterion, demonstrating that the smaller specimens behave more inelastic (plastic). On the other hand, in normal SCC, the failure tendency toward LEFM is attributed to the brittleness of this design, though with an increase in the steel fiber content, plastic failure occurs where a significant improvement in the energy absorption and ductility in compression is achieved by adding fibers to concrete [38]. In fact, steel fibers improve the bonding region by creating a strong bond between aggregate and matrix, preventing the crack propagation process, and increasing the specimen's ductility. This can be confirmed by the transitional size (D0), which increases as the fiber volume increases, where is in consistency with the results of Saeedian et al. 2017 [31], and Kazemi et al. 2017 [10]. The results correspond to both shapes of cube and cylinder. On the other hand, R2 values show the best result in designs with 0.3 of steel fiber content, since the specimens with fiber are more coherent. By comparing the results of SEL, it is observed that the tendency to be plastic in cylindrical specimens is greater than that of a cube specimen. Both fibrous and non-fibrous designs lead to large coefficients of D0 implying that the cylindrical specimens exhibit greater ductility than cubic specimens in laboratory tests, which can be due to the type of crack growth and ultimate failure. As stated previously, cubic specimens fail in the shear mechanism, while the cylindrical specimens are associated with the axial failure mode. Fig. 10 shows the compressive strength curve of specimens normalized to fcs versus the dimensions of specimens. In the base design, the size effect is the same for each of the two cubic and cylindrical shapes, however, once the steel fibers are added to the design, the size effect reduction in cylindrical specimens is more pronounced than in cubic ones with same dimensions. This indicates that the effect of steel fibers in a uniaxial compression failure better works than in shear failure- which occurs in the cube, and thus reduces the slope of the curve compared to base design. On the other hand, the compressive strength ratio index c, presented in Table 6, increases with the addition of steel fibers in both shapes, however, it is more obvious in cylinders, showing a greater decrease 57

Engineering Fracture Mechanics 206 (2019) 46–63

M. Akbari et al. 1.5 MSEL (Cu) MSEL(Cy)

fc(D)/fcs

1.3

SCC (Cu) SCC(Cy)

1.1 0.9 0.7 25

50

75

100

125

150

175

200

225

D (mm)

(a) 1.5

MSEL (Cu) MSEL(Cy)

fc(D)/fcs

1.3

SCC_0.1SF (Cu) SCC_0.1SF (Cy)

1.1 0.9 0.7 25

50

75

100

125

150

175

200

225

D (mm)

(b) 1.5 MSEL (Cu) MSEL(Cy)

fc(D)/fcs

1.3

SCC_0.2SF (Cu) SCC_0.2SF (Cy)

1.1 0.9 0.7 25

50

75

100

125

150

175

200

225

D (mm)

(c) MSEL (Cu)

1.5

MSEL(Cy)

fc(D)/fcs

1.3

SCC_0.3SF (Cu) SCC_0.3SF (Cy)

1.1 0.9 0.7 25

50

75

100

125

150

175

200

225

D (mm)

(d) Fig. 10. Comparison between cubes and cylinders specimen of modified size effect law (MSEL) (a):SCC (b):SCC-0.1SF (c): SCC-0.2SF (d): SCC-0.3SF.

in the size effect after the addition of steel fibers in cylindrical specimens. All R2 coefficients lie in an acceptable range, indicating that the experimental results and MSEL results are in good agreement. The constants of size effect obtained via fractal theory and presented in Table 6, express that the characteristic length has decreased with increasing compressive strength. This means that the structure tends to brittle response and its ductility is diminished, conforming to Asadollahi et al. 2016 [30]. Nevertheless, Fig. 11 signifies that by increasing the strength, failure tends to the Euclidian regime and as the strength decreases Fractal regime failure governs. In other words, increasing steel fiber content by 0.1, 0.2 and 0.3% in SCC, the specimen's ductility as well as its strength increase, though MFSL has not been able to capture the effect of fiber addition. Despite all coefficients of R2 = 0.99, are consistent with the results of this paper, MFSL was not capable of detecting the presence of steel fibers with the low percentage. In addition, adding fibers to both cubic and cylindrical specimens leads to the Euclidian regime failure. This is more pronounced in cylinders, specifying higher ductility of fibrous cylindrical specimens. In fact USEL is divided into two parts of descending and ascending sections, though compressive strength results show that only 58

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0.30 MFSL SCC SCC-0.1SF SCC-0.2SF SCC-0.3SF

Log (fc(D)/fcs)

0.25 0.20 0.15 0.10 0.05 0.00

Euclidian regime

-0.05 -0.3

-0.1

0.1

0.3

0.5

0.7

Log (D/Lch)

(a)

Log (fc(D)/fcs)

0.30

MFSL

0.25

SCC

0.20

SCC-0.1SF SCC-0.2SF

0.15

SCC-0.3SF

0.10 0.05 0.00

Euclidian regime -0.05 -0.3

-0.1

0.1

0.3

0.5

0.7

Log (D/Lch)

(b) Fig. 11. Multi fractal scaling law (MFSL) for (a):cubes (b):cylinders.

descending sections occurs. So as it was expected, the curve of the proposed model only captures the descending section. The initial results associated with the compressive strength, revealed that all the numbers decrease with increasing size of specimens. Thereby, the descending section can only be expected in the USEL model. Fig. 12 illustrates the trend of variation of compressive strength versus specimen size, which is well correlated as to the rest of the examined models. Table 7a also confirms the results of SEL, which increases with the increase in concrete ductility D0. It should be noted that D0 has the main effect on the compressive strength variation against size. On the other hand, since the maximum aggregate size as well as the aggregate type are constant, df, the fractal dimensions, which is directly related to the aggregate type and material, does not change significantly. It is worth mentioning that, a significant change in df, considerably alters the compressive strength versus specimen size diagram, specifically the amount of slope. Given the fact that the size effect model proposed by Masoumi et al. in 2015, is for rock and has very small dimensions, where specimens of this size are not practically possible to be constructed in concrete, unless the model is applied to mortar. Respecting such interpretations, since no decrease and increase in strength of small and large specimens were observed through the results of this paper, then the numbers obtained for diameter at integrate (di) are also unacceptable ones. Another reason in observing decrease and increase in the strength of rock specimens might be attributed to the integrity of the particles of rock specimens. Hence, the model can be applied to ultra high-strength concrete with high particle density, or to concrete maintenance in low temperatures which results in the coalescence of microcracks; and it is then possible to confirm the USEL model. Therefore, the authors suggest that the model should be adopted in concrete with different aggregates, or ultra high-strength concrete, otherwise USEL can only inspect the descending section in the conventional concrete. Table 7b depicts the results of MMFSL and IUSEL, which similar to USEL, have good agreements with the experimental results (R-squire). However, the descending section is only applicable for the current paper. In fact, the major difference between the IUSEL model vs. USEL, is its ability to estimate the size effect of specimens in a much wider range, which is of utmost importance. Yet, due to the relative similarity of the USEL and IUSEL diagrams, IUSEL curves have not been illustrated. In Fig. 13, curve pertaining to model of Sim et al. 2013 [26] is identical to the MSEL model, except that Sim's model considers more conditions such as concrete weight and the lateral depth of specimen for prediction. In this model, similar to MSEL, the size 59

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Compressive strength (MPa)

58

USEL (Cy) USEL (Cu)

53

SCC (Cy) SCC (Cu)

48 43 38 33 45

65

85

105

125

145

D (mm)

(a) USEL_0.1SF (Cy)

Compressive strength (MPa)

58

USEL_0.1SF (Cu) SCC _0.1SF (Cy)

54

SCC _0.1SF (Cu)

50 46 42 38 45

65

85

105

125

145

D (mm)

Compressive strength (MPa)

(b) USEL _0.2SF (Cy)

63

USEL_0.2SF (Cu) SCC-0.2SF (Cy)

58

SCC-0.2SF (Cu)

53 48 43 38 45

65

85

105

125

145

D (mm)

(c) Compressive strength (MPa)

65

USEL_0.3SF (Cy) USEL_0.3SF (Cu)

60

SCC_0.3SF (Cy) SCC_0.3SF (Cu)

55 50 45 40 45

65

85

105

125

145

D (mm)

(d) Fig. 12. Comparison between cubic and cylinder specimens of USEL (a):SCC (b):SCC-0.1SF (c): SCC-0.2SF (d): SCC-0.3SF.

effect of cylindrical specimens is of lower significance than cubic ones. In small specimens, the size effect is larger, and as the specimen size grows, it decreases and both shapes curves become roughly tangent. Also, by adding steel fibers, the size effect reduction is greater in cylindrical specimens compared to cubic specimens. Table 6 reports the constants of Sim et al.’s 2013 [26] relation along with R2 values. It is worth mentioning that R2 values are greater in cubes as to cylinders. The Sim et al.’s 2013 [26] relation is proposed for lightweight concrete, however, it is proved that it can be used for SCC and is advantageous in predicting the size effect, especially for cubic specimens. Concerning the results obtained from models, the best model for estimating the size effect in the cylindrical and cubic specimens was the MFSL, giving greater R2 values, even though it is unable to identify the low fiber contents. The results of all cubic specimens 60

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Table 7 results of regression analyses. (a):USEL (b):IUSEL (a):USEL Mix design

Sample

Bfcs

D0

σ0

df

di

R2

SCC

Cy Cu

52.786 66.564

130.540 82.696

52.849 66.585

0.999486 0.999855

100.653 86.005

0.98 0.99

SCC_0.1SF

Cy Cu

51.111 68.722

238.134 90.544

51.196 68.704

0.999270 1.000120

95.888 93.293

0.97 0.99

SCC_0.2SF

Cy Cu

52.449 71.113

247.526 100.178

52.466 71.178

0.999858 0.999600

104.053 99.803

0.83 0.98

SCC_0.3SF

Cy Cu

53.366 76.191

254.234 90.308

53.360 76.185

1.000046 1.000030

112.793 146.590

0.99 0.99

(b):IUSEL Mix design

Sample

fm

df

di

Maximum strength at di

R2

SCC

Cy Cu

28.840 34.196

0.998986 0.982993

83.089 81.560

41.160 46.303

0.99 0.95

SCC_0.1SF

Cy Cu

35.241 37.166

0.997752 0.980533

98.128 85.811

42.672 48.273

0.99 0.93

SCC_0.2SF

Cy Cu

36.776 39.019

0.996276 0.990500

103.140 58.888

43.697 51.008

0.98 0.94

SCC_0.3SF

Cy Cu

37.789 39.866

0.995903 0.990753

106.570 98.205

44.434 51.864

0.98 0.98

are slightly better than those of cylinders, connoting that the above models (SEL, MSEL, MFSL, Sim et al. [26]) are also suitable for SCC and SFRSCC cubic specimens. Furthermore, all models demonstrate size effect reduction by increasing the steel fiber content. SCC-0.3SF has the highest coefficient of R2 in almost all results, which indicates the effect of steel fibers on microcracks and their growth hindrance. Besides, SEL is the best model for better understanding of brittleness and ductility, which in addition to estimating the size effect, it also examines the behavior of concrete and specifies the effect of fiber. According to the output of various size effect models, it is noteworthy that cylindrical specimens demonstrated more ductility as well as less variety in results compared to cubic specimens. 5. Conclusions The present article aimed to investigate the effect of steel fibers on the compressive strength size and shape effects which depend on the material size effect property. Different size and shape effect models were examined and the coefficients were modified for the SCC and SFRSCC. Moreover, the relations between the conventional cylindrical and cubic specimens were established. The following results are to be noted. 1. The compressive strength of cubic specimen is obtained greater than those of cylindrical ones. In addition, the nominal compressive strength decreased with increasing dimensions. By adding 0.1, 0.2 and 0.3% of steel fiber to SCC, compressive strength increased slightly. Although steel fibers reduced the tested properties of fresh SCC, the percentages used for SCC designs were still reasonable. 2. The results of Neville 1986, showed normal SCC and SFRSCC were unable to reduce the shape effect. Furthermore, in studying the effect of shape, specimens with 100 mm cross sections had the best agreement among the compressive strength of four designs. Subsequently, this size is suggested for use in projects regarding this type of concrete. In addition, some relations between cubes and cylinders were presented compared to the standard cylinder specimen. 3. The size effect in cubes was greater than that of the cylinder; this also applies to SFRSCC concrete. By increasing the volume fraction of steel fibers, the size effect decreased, which is due to crack bridging and the reduction of crack growth. Therefore, the use of a cylindrical specimen is recommended. 4. The results of SEL and MFSL showed large specimens have greater brittleness, thus becoming more ductile as the specimen dimensions decrease. This is more apparent in cylindrical specimens. By adding and increasing the steel fiber content, failure moves towards the strength criteria region in the SEL and towards the Euclidian regime in the MFSL, indicating a decrease in brittleness and an increase in ductility. 5. Coefficients of shape and size effects on the compressive strength, as a concrete property, were modified in the models Kim and Yi, Neville, SEL, MSEL, MFSL and Sim et al. for NSCC and SFRSCC. The obtained values were in good agreement with those of experimental results being convenient for estimating size effect.

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1.5

SCC (Cu) SCC (Cy)

fc(D)/fcs

1.3

Sim et al. (Cu) Sim et al. (Cu)

1.1 0.9 0.7 25

75

125

175

225

D (mm)

(a) 1.5

SCC_0.1SF (Cu) SCC_0.1SF (Cy)

fc(D)/fcs

1.3

Sim et al. (Cu) Sim et al. (Cy)

1.1 0.9 0.7 25

75

125

175

225

D (mm)

(b) 1.5

SCC_0.2SF (Cu) SCC_0.2SF (Cy)

fc(D)/fcs

1.3

Sim et al. (Cu) Sim et al. (Cy)

1.1 0.9 0.7 25

75

125

175

225

D (mm)

(c) 1.5

SCC_0.3SF (Cu) SCC_0.3SF (Cy)

fc(D)/fcs

1.3

Sim et al. (cu) Sim et al. (Cy)

1.1 0.9 0.7 25

75

125

175

225

D (mm)

(d) Fig. 13. Comparison between cubes and cylinders specimen of Sim et al. model (a):SCC (b):SCC-0.1SF (c): SCC-0.2SF (d): SCC-0.3SF.

6. The results of size effect showed that cubic specimens also have the potential to predict the abovementioned size effects, showing more consistency as to cylindrical specimens in some cases. MFSL had the best agreement, though unable of detecting low percentages of steel fibers. Additionally, the highest value of R2 is associated with SCC of 0.3 steel fiber content, suggesting that steel fibers improve the bond in concrete providing a more coherent concrete and thereby reducing the size effect.

Acknowledgement Authors would like to appreciate the academic members of the Faculty of Civil Engineering in the Babol Noshirvani University of Technology of Iran who kindly examined the research and suggested useful comments and modifications.

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