Steel fibre self-compacting concrete under biaxial loading

Construction and Building Materials 224 (2019) 255–265

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Steel fibre self-compacting concrete under biaxial loading R.N. Mohamed a,⇑, N.F. Zamri a, K.S. Elliott b, A.B.A. Rahman a, N. Bakhary a a b

School of Civil Engineering, Faculty of Engineering, Universiti Teknologi Malaysia, Johor Bahru, Johor 81310, Malaysia Department of Civil Engineering, Faculty of Engineering, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

h i g h l i g h t s
Plain and fibrous concrete were tested under three multiaxial stress state.
Strength and behaviour of concrete were analysed with respect to the stress ratio.
Failure modes of concrete under uniaxial and multiaxial stress state were compared.
Failure criterions of concrete under multiaxial stress state were proposed.

a r t i c l e

i n f o

Article history: Received 3 January 2019 Received in revised form 7 July 2019 Accepted 11 July 2019

Keywords: Biaxial strength Uniaxial strength Steel fibre Steel fibre self-compacting concrete

a b s t r a c t This study involves the investigation of the effects of steel fibre content volume on the biaxial stress behaviour of steel fibre self-compacting concrete (SFSCC) at different stress ratios. This study covers compression-compression, compression-tension, and tension-tension stress regions. The results are discussed in terms of the uniaxial and biaxial strength, stress-strain relationship, and failure mode of SFSCC specimens. In terms of strength, 1.0% fibre volume fractions showed the highest increment in biaxial compression and compression-tension, which were 55% and 84%, respectively, when compared to plain concrete. This improvement was due to the integration of steel fibre. In contrast with compression strength, biaxial tension strength decreased in comparison to uniaxial tensile strength. Additionally, based on the octahedral stress space, the failure criteria of concrete for each region were proposed in a quadratic polynomial equation, and the parameters were derived from a regression analysis. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Fibre reinforced concrete is utilized for various applications, including bridges, buildings, and tunnels. Over the past several decades, considerable efforts have been dedicated to studying the influence of steel fibres in concrete subjected to uniaxial loading [1–5]. Moreover, the use of self-compacting concrete as a medium for transporting steel fibre has considerably highlighted the benefits of fibre reinforced concrete. The randomness of steel fibre in a concrete matrix has been proven to noticeably enhance the flexural and splitting tensile strength [6–8] and toughness of concrete [9] by stitching the gap between two sides of a crack’s opening [1,10]. In real structural applications, there are almost no reinforced concrete structures that carry only uniaxial stress. Most of reinforced concrete structures exposed to a multiaxial stress state. However, the employed analysis and design methods are still based on material properties obtained from the basic uniaxial ⇑ Corresponding author. E-mail address: [email protected] (R.N. Mohamed). https://doi.org/10.1016/j.conbuildmat.2019.07.076 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.

strength test. This trend has resulted in underestimated and overestimated designs which are not economical and might be unsafe. For instance, if the capacity of concrete in multiaxial compressiontension region (C-T) is overestimated, this would lead to failure in structural application. Many studies have examined the strength and deformation of steel fibre reinforced concrete (SFRC) subjected to biaxial loading. The results of these studies have been inconsistent. Most of the studies have limited their interest to biaxial compression [11– 16], and only few studies have concentrated on the behaviour of concrete under biaxial tension. For example, Bao et al. (2018) [11] conducted a series of biaxial compression tests on SFRC. The findings regarding the strain rate and stress ratio showed a great impact on the failure mode of SFRC. In terms of strength, the results corroborated by Lim and Nawy (2005) [12] and Shang and Ji (2014) [17], showed the biaxial compressive strength was always greater than the uniaxial compressive condition. The ultimate biaxial strength of concrete under biaxial tension–compression is less than that of both uniaxial compressive and tensile strength, as reported by Shang et al. (2014) [18], Wang and Song

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(2008) [19], and Hussein and Marzouk (2000) [20]. Abdull-Ahad (1989) [21] has carried out many studies investigating SFRC under biaxial stress involving compression-compression (C-C), C-T, and tension-tension (T-T) regions. It was found that biaxial stress increases the strength of SFRC specimens under biaxial compression and biaxial tension by about 38% and 79%, respectively, at sequential stress ratios of 0.5 with 1% steel fibre volume. However, no failure criterion of concrete has been proposed for use in a full range of biaxial studies on SFRC or SFSCC. Apart from differences in strength and deformation behaviour, the failure criterion of a concrete matrix subjected to biaxial stresses also differs from that of a concrete matrix subjected to uniaxial stress. With the development of numerical analyses in concrete structures, the importance of understanding the behaviour of concrete subjected to biaxial loading cannot be ignored. The failure envelope produced in studies on fibrous concrete play a significant role in stimulating fractures and causing failure in concrete. Therefore, many researchers have proposed the failure criteria of concrete under biaxial loading [22,17]. These criteria can be summarized into three categories of derivation formulas: formulas based on classical theories, regressive calculations of the experimental data, and failure envelope characteristics. Unfortunately, the mathematical equation and failure envelope derived from classical theories of material (e.g., the Mohr-Coulomb failure criterion and the von Mises failure criterion) are too simple and inaccurate to be used for concrete, as they consider the influence only of first principal shear stress. According to the authors’ literature review, many researchers have studied the biaxial behaviour of fibre concrete. However, only a few studies have concentrated on the full range of biaxial stresses regions (C-C, C-T, and T-T) specifically on steel fibre selfcompacting concrete (SFSCC). As SFSCC is still considered as new technology in concrete applications, the requirement on its uniaxial and biaxial failure envelopes is seen as an essential criterion in concrete analysis. Therefore, in this study, experiments have been carried out in which attempts have been made to gain insight into the actual behaviour of the elements subjected to multiaxial forces. Three biaxial stresses region were considered, namely compression-compression (C-C), compression-tension (C-T), and tension-tension (T-T), tested under three different stress ratios. This experimental study involves tests on three types of concrete mix design, comprising 108 specimens. Based on the test results, a biaxial failure envelope curve was developed, and a failure criterion was suggested. 2. Experimental works 2.1. Concrete mix design Ordinary Portland cement (grade CEM 1 42.5R) with a specific gravity of 3.14 was used in the concrete mix in the present investigation. Local river sand and well-graded 10-mm rounded coarse aggregates (which comply with BS 882 (1992) [23]) were used as the fine and coarse aggregates, respectively. The typical relative densities (SSD) of the fine and coarse aggregates were 2.68 and 2.66, and their typical absorption values were 1.15% and 1.0%, respectively. In this study, high powder content was required to ensure the workability of the concrete while limiting the risk of segregation and reducing heat generation. These requirements were met via the replacement of about 24–26% of the cement content with class C fly ash with a specific gravity of about 2.1, as suggested in previous studies [24,25]. Three volume fractions (vf = 0%, 0.5%, and 1%) of Dramix hooked-

end section circular steel fibres (RC 65/35 BN type, length = 35 mm, aspect ratio = 65, density = 7850 kg/m3) were incorporated into the concrete mix as shown in Fig. 1. Hooked-end steel fibre was used because it has been reported to promote acceptable bond behaviour between the fibre and concrete matrix. Furthermore, the bond depends not only on chemical adhesion and static friction but also on the mechanical interlocks developed by the hooks of the fibres [26,27]. Table 1 shows the material compositions of the concrete mixes. For selfcompacting concrete, self-compactability characteristics can be achieved only by reconciling the opposing effects of deformability and viscosity. With the inclusion of steel fibres, which is known to restrict workability, slight modifications have been considered to accommodate the steel fibres without reducing self-compactability. Therefore, the mix designs of SFSCC 0.5 and SFSCC 1.0 were developed based on a normal SCC mix. For the SFSCC 0.5 mix, 0.5% steel fibre was incorporated; for the SFSCC 1.0 mix, 1.0% steel fibre was incorporated. A superplasticizer based on the soluble salt of a polymeric naphthalene sulphonate Daracem SP1 (GRACE Construction) was used to balance the workability of the SFSCC mixes. The rheological properties of the SCC and SFSCC mixes were accessed through slump flow, L-box, and V-funnel tests in accordance to European guidelines [28]. The results of these tests are presented in Table 2. The results complied with criteria related to self-compactability characteristics put forth by European guidelines [28]. Thus, the mixes were rightfully classified as self-compacting concrete mixes. In addition, the results revealed the similarities between the mixes used in this study and those with the same amount of steel fibre volume fractions mentioned in the existing literature in terms of flowability and filling ability characteristics [29,30]. 2.2. Specimen details The dimensions and shapes of the examined specimens vary depending on the mode of applied stress. Uniaxial and biaxial compression specimens were of a plate size of 100 mm  100 mm  50 mm (see Fig. 2), and C-T specimens were cast in 100 mm  400 mm  50 mm prisms. To allow for the application of tension forces, all C-T specimens were cast with two high-tensile threaded reinforcing bars (diameter = 10 mm) embedded at two vertical ends at 150 mm (Fig. 3(a)). Lastly, the T-T specimens were cast in a cross-shaped mould with dimensions of 450 mm  450 mm  100 mm. All T-T specimens were also embedded with two high-tensile threaded reinforcing bars (diameter = 10 mm) at four vertical ends (Fig. 3(b)). 2.3. Test procedures During the tests, the major principal stress, f1, and the minor principal stress, f2, were applied simultaneously at a loading rate of 0.2 MPa/s. Proportional loading was used for all tests, and the ratio of the two applied loads (f1/f2) was kept constant throughout the tests. Three different sets of stress ratios (0.2, 0.5, and 1.0), (0.05, 0.1, 0.2, and 0.4), and (1.0, 2.0, and 5.0) were used for the C-C, C-T, and T-T specimens, respectively (Fig. 4). The C-C specimens were installed between the loading plates as shown in Fig. 5 (a). A 2-mm clearance area was provided at each side to prevent the loading platens from contacting each other during the tests. For both vertical and horizontal directions, 0.5-mm thick polyethene friction-reducing pads (PTFE sheets) made of polyethene were placed between the loading platen and specimens to reduce the lateral restraint (Fig. 5(b)). These pads were noted by previous researchers as being effective in minimising the friction between loading platens and specimens [31,14,32]. Regarding the C-T and T-T test series, as shown in Fig. 6(a) and 6(b), tension forces were applied at the two rods, which were embedded at the ends of the specimens, thus allocating the free friction as per the C-C test. In general, the two rods at one end were forced (in a pulling direction) using a 250-kN hydraulic jack, while the other end remained fixed. Table 1 Material composition of concrete mixes. Concrete mix

SCC

SFSCC 0.5

SFSCC 1.0

Cement (kg/m3) Fly ash (kg/m3) Water (kg/m3) Coarse aggregate (kg/m3) Fine aggregates (kg/m3) Steel fibre (kg/m3)

406 132 172 740 833 0

412 132 168 740 833 39

422 132 170 740 833 79

Fig. 1. Dramix hooked end section of circular steel fibre.

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R.N. Mohamed et al. / Construction and Building Materials 224 (2019) 255–265 Table 2 The rheological properties of concrete mixes. Parameter

SCC

SFSCC 0.5

SFSCC 1.0

Slump flow (mm) L-Box, (H2/H1) ratio V-Funnel (s)

680 0.95 10

680 0.86 8

680 0.89 9

Requirement limit by EFNARC [28] Minimum

Maximum

550 0.80 6

850 1.0 12

Fig. 2. Specimen details: C-C specimen.

Fig. 3. Specimen details: (a) C-T specimen and (b) T-T specimen. To ensure the specimen was perfectly aligned and fixed vertically and horizontally, a small load was applied, and the readings made by a strain gauge were checked. Afterwards, the load was progressively increased until failure occurred. The ultimate failure loads were recorded by a data-acquisition system. Two strain gauges were mounted in an orthogonal direction on the centre surface of each of the specimens to measure deformation.

3. Results and discussion 3.1. Uniaxial and biaxial strength Failure strength envelopes were plotted based on the average failure loads of all C-C, C-T, and T-T specimens (Fig. 7). All points

were connected in a smooth, convex curve following the most widely used biaxial failure envelopes in concrete proposed by Kupfer-Gerstle. Each point in the graph represents an average strength value of at least three specimens. Summaries of the uniaxial and biaxial strength of each concrete specimen’s test results are presented in Table 3. For the C-C specimens, the confinement stress in the minor direction, f2, has a pronounced effect on the specimens’ strength. Also, the strength of C-C specimens under biaxial compression was higher than that of C-C specimens under uniaxial compression. The increments of biaxial strength over uniaxial strength for SCC, SFSCC 0.5, and SFSCC 1.0 were 21%, 39%, and 64%, respectively. The greatest increase observed when comparing specimens under

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Fig. 4. Stress ratio, f2/f1, used in the experimental works.

Fig. 5. (a) C-C test setup and (b) the use of compression steel platens and friction reducing pads in C-C tests.

uniaxial stress to specimens under biaxial compression was attained at the stress ratio of f2/f1 = 0.5. This result supports the trend reported by Bao et al. (2018) [11], Shang and Ji (2014) [17], Traina (1991) [14], and Abdull-Ahad (1989) [21] in normal SFRC. It also supports the results reported by Hussein and Marzouk (2000) [20] and Lim and Nawy (2005) [12], whose studies involved high-strength concrete. The addition of steel fibre to SCC increased the ultimate strength for both uniaxial and biaxial strength. The inclusion of 0.5% and 1.0% steel fibre increased uniaxial strength by about 8%

and 14%, respectively. The strength increase was more noticeable under biaxial loading, where the inclusion of 0.5% and 1.0% steel fibre contributed to increases in biaxial strength of up to 24% and 55%, respectively. Again, all values were attained at a stress ratio of 0.5. For SFSCC 0.5, the addition of steel fibre did not influence the strength up to f2/f1 = 0.2 due to mismatched action between the fibre content and force pressure. The fibres in such a stressstate had just experienced relief before they started to resist the load. As a result, it can be concluded that the effect of steel fibres in compression varies depending on the test conditions. Shang

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Fig. 6. (a) C-T and (b) T-T test setups.

Compression - tension region

10

Tension - tension region

0 -100

-80

-60

-40

-20

-10

0

20

-20 f2 (N/mm2)

-30 -40 -50

SCC SFSCC 0.5 SFSCC 1.0

-60 -70 -80 -90

Compression - compression region

-100

f1 (N/mm2)

Fig. 7. Failure strength envelopes of concrete.

and Ji (2014) [17] also reported that many factors affect the mechanical behaviour of concrete under biaxial compression, including the concrete type, loading rate, and testing machine used. In contrast to the biaxial behaviour observed in C-C specimens, the presence of compressive stress, f2, in C-T specimens proportionally decreases tensile strength along the vertical axis. The compressive strength at failure is inversely proportionate to the acting tensile strength, and the variation depends on the stress ratio. The inclusion of 1.0% steel fibre increased biaxial strength by up to 84%

at a stress ratio of 0.05 when compared to plain SCC specimens. When the stress ratio was increased above 0.05, the strength of SFSCC decreased dramatically. When comparing the curve patterns between the SCC and SFSCC specimens, it was observed that the incorporation of steel fibres results in a smoother biaxial stress curve. Meanwhile, in T-T specimens, the ultimate strength under biaxial strength is less than it is under uniaxial strength. This value depends on the stress ratio applied. At a ratio of f1/f2 = 1, the differences between uniaxial and equivalent biaxial strength are about 12%, 30%, and 25% for plain SCC, SFSCC 0.5, and SFSCC 1.0, respectively. In this case, the stress ratio of biaxial loading is directly proportionate to the strength of SFSCC. When the stress ratio of biaxial loading is progressively decreased, the strength of SFSCC also decreases significantly. This trend continues until the minimum strength values are reached at equal biaxial tension (f1/f2 = 1.0). This is in line with the findings of Rosenthal et al. (1970) [33] and Kwak and Kim (2006) [34], who found that biaxial tensile strength decreases in proportion to uniaxial tensile strength. With the inclusion of 0.5% and 1.0% steel fibre, ultimate tensile uniaxial strength increased by about 17% and 26%, respectively. When the stress ratio of biaxial loading progressively decreased, (i.e., when it changed from uniaxial to biaxial), the strength increments also uniformly decreased due to the inclusion of steel fibres. Eventually, minimum strength values were reached at equal biaxial tension (f2/f1 = 1.0). The highest increment was attained by the uniaxial tensile specimen, and the increment was within the range of expected values of improvements in tensile strength (i.e., 25% to 30%).

Table 3 Experimental stress ratios of uniaxial and biaxial strength. Region

Compression-compression (C-C)

Compression-tension (C-T)

Tension- tension (T-T)

Stress ratio

0 1.0 0.5 0.2 0.05 0.1 0.2 0.3 0 5.0 2.0 1.0

SCC

SFSCC 0.5

SFSCC 1.0

fcu = 45 N/mm2

fcu = 50 N/mm2

fc = 56.5 N/mm2

f1/fcu

f2/fcu

f1/fcu

f2/fcu

f1/fcu

f2/fcu

0 1.108 0.596 0.218 0.027 0.036 0.049 0.062 0.075 0.074 0.074 0.067

1.00 1.113 1.189 1.088 0.549 0.364 0.239 0.172 0.000 0.015 0.037 0.067

0 1.234 0.684 0.226 0.032 0.037 0.050 0.057 0.079 0.076 0.076 0.060

1.00 1.244 1.362 1.118 0.640 0.372 0.248 0.193 0.000 0.015 0.038 0.060

0 1.341 0.761 0.263 0.039 0.042 0.048 0.053 0.071 0.067 0.064 0.057

1.00 1.342 1.522 1.308 0.777 0.428 0.243 0.178 0.000 0.014 0.033 0.057

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more prominent in the SCC samples than in the SFSCC specimens. Similar observations were noted by Shiming and Yupu (2013) [37], Lee et al. (2004) [31], and Hussein and Marzouk (2000) [20]. This effect is caused by the fact that principal tensile stresses in the orthogonal direction increase as the absolute value of f2/f1 increases, thereby decreasing the principal compressive stress. The SFSCC specimens showed a better failure definition than the SCC specimens. The SFSCC specimens achieved a compressive failure strain of 3000 me, whereas the compressive failure strain of SCC was about 1250 me. The failure strains for T-T specimens were quite small; none exceeded 100 me, and the individual values were very close to each other. A similar range of strain values was obtained by Tasuji et al. (1978) [38]. Hussein and Marzouk (2000) [20] concluded that the magnitude of the tensile strain at failure is not constant or significant for each individual mix under different load combinations. In stress-strain relationships, the proportional limit can be defined as the point at which the stress-strain curve deviates from linearity. At all stress ratios for uniaxial and biaxial compression, the stress-strain curves for SFSCC 1.0 specimens exhibited linear behaviour up to a higher level of stress than SCC specimens did. Also, the SFSCC 1.0 specimens showed more linear behaviour than the corresponding SCC specimens. The best explanation for this is that as the minor principal stress, f2, is increased, the proportional limit is also increased. Internal microcracking is a major cause of

3.2. Stress-strain relationship The principal strains, e1 and e2, fluctuate in the direction of principal stresses, f1 and f2, respectively. This relationship is such that e1 > e2 when the tensile strain is positive. The stress-strain relationships for SCC and SRSCC 1.0 obtained from the C-C, C-T, and T-T tests are shown in Figs. 8, 9, and 10, respectively. The results for SFSCC 0.5 are not included, as they were not remarkably different from those for SCC. Due to the confinement effect in biaxial compression (C-C test), when minor principal compressive stress, f2, was introduced, the point at which the stress-strain curve began to deviate from linearity increased. This phenomenon was observed in all C-C specimens. Furthermore, as reported by Lee et al. (2004) [31], Seow and Swaddhiwudhipong (2005) [35], and Foltz et al. (2017) [16], when the increased confinement from biaxial compression had a pronounced effect on strength and deformational behaviour, stiffness and ultimate strength increased. For SFSCC 1.0, the maximum principal compressive and tensile strains were about 2545 me and + 1250 me, respectively. For SCC, these values were about 2025 me and + 662 me. These failure strains are slightly lower than the values proposed in the code of practice [36] (i.e., between 3000 and 3500 me). In the C-T test, the principal failure for compressive and tensile strain decreased as tensile load increased. This phenomenon was

100

100

Applied Stress (N/mm2)

Applied Stress (N/mm 2)

80 SCC 60

40 0.2

40 0.5 20

0

1

1

0 0 -3000

SFSCC-1.0 60

0.2

20

0.5

80

-2500

-2000

-1500

-1000

-500

0

500

-3000

1000

-2500

0 -2000

-1500

-1000

-500

0

500

1000

Strain (με)

Strain (με) Fig. 8. Stress-strain relationships of SCC and SFSCC 1.0 for C-C.

40

40

Applied stress (N/mm2)

Applied stress (N/mm2)

35 30 SCC 20

-0.05

10

-0.1 -0.2

-3000

25 20

10

0.2/-1

-1000

Strain (με)

5

0.1/-1

0 -2000

SFSCC-1.0

15 0.05/-1 0.4/-1

-0.4 -4000

30

0

1000

2000

0 -4000

-3000

-2000

-1000

Strain (με) Fig. 9. Stress-strain relationships of SCC and SFSCC 1.0 for C-T.

0

1000

2000

261

5

5

4

4 Applied stress (N/mm2)

Applied stress (N/mm2)

R.N. Mohamed et al. / Construction and Building Materials 224 (2019) 255–265

SCC 3

2 1 2

1

0

3

2 1 5 0

0 -50

SFSCC-1.0

1

2

5 -25

0

25

50

75

100

0 -50

-25

Strain (με)

0

25

50

75

100

Strain (με) Fig. 10. Stress-strain relationships of SCC and SFSCC 1.0 for T-T.

non-linearity of the stress-strain curve of concrete. Microcracks begin at the matrix interface and propagate throughout the mortar matrix, eventually causing failure. The presence of minor stress, f2, prevents these microcracks from propagating, thus resulting in stiffer and more linear stress-strain responses in the major principal direction. 3.3. Failure mode Fig. 11 shows the crack patterns for SCC and SFSCC 1.0 subjected to biaxial compression. For both SCC and SFSCC 1.0 under uniaxial compression, splitting type cracks were prevented from forming when biaxial minor principal stress was introduced. This phenomenon indicates that minor principal stress, f2, can prevent the occurrence of tensile stress caused by major principal stress, f1. Surface fractures occurred along a plane parallel to the test specimen. Almost all failures in SCC samples produced a loud blast as the specimens were crushed into small pieces. In contrast to the uniaxial failure mode, SFSCC 1.0 specimens under biaxial compression showed shear type failure. Cracks formed on an inclined shear

plane, varying between 15° and 40° depending on the stress ratio and fibre volume concentration. The results of the comparison between SCC and SFSCC 1.0 specimens show that the confinement stress along the minor principal direction, f2, can change the failure mode from splitting tensile failure to shear failure. Typical C-T crack patterns are shown in Fig. 12(a). For SCC specimens, the fracture was explosive. One large crack appeared abruptly, and the specimen divided into two pieces. With the inclusion of steel fibres, failure occurred through a pull-out process and was accompanied by a cracked region that was either smaller or larger than the cracked region in non-fibrous specimens. The size of the cracked region depended on the fibre distribution content and tensile resistance of the specimen after cracking began. Under uniaxial tension, failure occurred due to the formation of a single crack that ran perpendicular to the direction of loading and parallel to the plane of the specimen. Biaxial tension resulted in a similar type of failure crack, which ran perpendicular to the direction of the greater tensile force (see Fig. 12(b) and 12(c)). In the case of equal biaxial tension, there was no preferred direction for the fracture surface, and the cracks always ran parallel to the

Fig. 11. Crack patterns for (a) SCC and (b) SFSCC 1.0 under biaxial compression.

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Fig. 12. Crack patterns for (a) SCC and SFSCC 1.0 in a C-T test, (b) SCC under uniaxial tension, and (c) SFSCC 1.0 in a T-T test.

unloaded surface. For SFSCC 1.0 specimens under uniaxial and biaxial tension, an observation was made similar to that made for C-T specimens (i.e., to ensure that failure occurred due to the pull-out of steel fibres before yielding). The failure was of a ductile fashion, with several cracks propagating before ultimate failure took place.

3.4. Biaxial strength envelopes diagram From the test results for C-C, C-T and T-T, the ultimate strength data were normalised with respect to the unconfined uniaxial 0.2 0.0 -2.0

-1.5

-1.0

-0.5

-0.2

0.0

0.5

compressive strength, fcu, required to produce stress envelopes which covered all regions. These data are summarized in Table 3. The average unconfined compressive strength values for SCC, SFSCC 0.5, and SFSCC 1.0 obtained from control tests were 45, 49, and 56 N/mm2, respectively. Fig. 13 shows the stress envelopes based on normalised strengths f1/fcu and f2/fcu. A comparison of the SCC and SFSCC curves reveals an increase in strength as steel fibre content was increased from 0% to 1%. Steel fibres appear to improve the properties of plain concrete under biaxial stress (in C-C and C-T regions) and under uniaxial tension (in the T-T region). The confinement effect imposed by the fibres due to their pull-out resistance ensures proper ductility behaviour. According to Lim and Nawy (2005) [12], this advantage reduces the number of lateral steel reinforcements required in the production of concrete. In the T-T region, tensile strength decreased slightly when the two principal tensile stresses were equal.

-0.4 -0.6 -0.8 -1.0

SCC SFSCC 0.5 SFSCC 1.0

-1.2 -1.4 -1.6 -1.8

f1/fcu

3.5. Failure criterion of concrete The failure criterion of concrete is an expression of a mathematical function that describes the spatial failure envelopes of concrete. It is developed by summarizing many experimental data and can be used in any study to which it is relevant. Based on the octahedral stress space, three principal stresses can be expressed via the following equations:

1 3

roct ¼ ðr1 þ r2 þ r3 Þ

Fig. 13. Normalised stress envelopes.

ð1Þ

Table 4 The values of normal stress and shear stress in an octahedral stress space. Region

Compression-compression (C-C)

Compression-tension (C-T)

Tension- tension (T-T)

Stress ratio

0 1.0 0.5 0.2 0.05 0.1 0.2 0.3 0 5.0 2.0 1.0

SCC

SFSCC 0.5

SFSCC 1.0

fc = 45 N/mm2

fc = 50 N/mm2

fc = 56.5 N/mm2

roct /fc

soct /fc

roct /fc

soct

roct

soct

0.33 0.74 0.59 0.43 0.17 0.11 0.06 0.04 0.02 0.03 0.04 0.04

0.47 0.52 0.48 0.47 0.26 0.18 0.13 0.10 0.04 0.03 0.03 0.03

0.33 0.83 0.68 0.45 0.20 0.11 0.07 0.05 0.03 0.03 0.04 0.04

0.47 0.59 0.56 0.48 0.31 0.19 0.13 0.11 0.04 0.03 0.03 0.03

0.33 0.89 0.76 0.52 0.25 0.13 0.07 0.04 0.02 0.03 0.03 0.04

0.47 0.63 0.62 0.57 0.38 0.21 0.13 0.10 0.03 0.03 0.03 0.03

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-0.4 0

0.2

0.4

0.6

0.8

soct

1

-0.45

SCC SFSCC 0.5 SFSCC 1.00 Poly. (SCC) Poly. (SFSCC 0.5) Poly. (SFSCC 1.00)

/fc

-0.6 -0.65

/fc

Fig. 14. The failure criterion of concrete under biaxial compression based on octahedral stress space.

soct fc

0 -0.1

-0.1

0

0.1

0.2

0.3

/fc

-0.3

-0.5 -0.6

¼f þg

roct fc

roct

þ hð

fc

Þ

2

ð3Þ

According to previous research, quadratic parabola functions are preferable for describing the failure criterion of concrete under multiaxial stress due to high axial compression [39]. Through datafitting, the failure criterion of SFRC can provide the basis for analysing the performance of SFRC structures under multiaxial stress. For example, Equation (3) can be used to analyse concrete with 1.0% steel fibre under different conditions through the substitution of the regression coefficients as follows: Compression-compression:

0.4

-0.2

-0.4

ð2Þ

Eqs. (1) and (2) are used to calculate the values of normal stress,roct , and shear stress,soct . These values are then divided by the uniaxial compressive strength value of concrete (Table 4). Based on a mathematical regression analysis, the proposed quadratic parabola functions (Equation (3)) can adequately represent the failure criterion of concrete for each region (Figs. 14–16). The values of f, g, and h are summarized in Tables 5, 6, and 7 for compression-compression, compression-tension, and tensiontension regions, respectively. Also, the fitting results obtained using a quadratic parabola function were satisfactory in terms of the correlation coefficient, R2, value.

-0.5 -0.55

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr1  r2 Þ2 þ ðr2  r3 Þ2 þ ðr3  r1 Þ2 ¼ 3

SCC SFSCC 0.5 SFSCC 1.00 Poly. (SCC) Poly. (SFSCC 0.5) Poly. (SFSCC 1.00) /fc

soct fc

¼ 0:2043  1:0058 rfoct þ 0:5957ðrfoct Þ c

2

c

Compression-tension: sfoct ¼ 0:0504  1:2239 rfoct  0:1971ðrfoct Þ c

c

2

c

Tension-tension:sfoctc ¼ 0:1080  4:8122 rfoct  70:505ðrfoct Þ c c

2

Fig. 15. The failure criterion of concrete under biaxial tension-compression based on octahedral stress space.

4. Conclusions -0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.02 -0.005 0

The following conclusions were reached from the present investigation concerning the behaviour of SFSCC under different types of biaxial stress with different stress ratios:

-0.025 SCC SFSCC 0.5 -0.03 SFSCC 1.00 Poly. (SCC) -0.035 Poly. (SFSCC 0.5) Poly. (SFSCC 1.00) -0.04

/fc

-0.045

/fc

1. The ultimate strength of concrete under biaxial compression is greater than it is under uniaxial compression. Oppositely, the tensile strength of concrete is greater under uniaxial tension than under biaxial tension. Under biaxial compression at a stress ratio of 0.5, the highest increases in strength were about 24% and 55% for SFSCC with the addition of 0.5% and 1.0% of steel fibre volume content, respectively. The maximum increase in biaxial strength for SFSCC C-T specimens when compared to plain SCC was 84%. This increase was recorded at a stress ratio

-0.045

Fig. 16. The failure criterion of concrete under biaxial tension based on octahedral stress space.

Table 5 The regression coefficients for compression-compression. Types of concrete mix

SCC SFSCC 0.5 SFSCC 1.0

Fibre volume fraction, vf (%)

0 0.5 1.0

The fitting parameters f

g

h

R2

0.5554 0.4046 0.2043

0.4211 0.1499 1.0058

0.5031 0.0944 0.5957

0.9921 0.9743 0.9973

f

g

h

R2

0.0601 0.0396 0.0504

1.0679 1.0468 1.2239

0.5323 0.8826 0.1971

0.9995 0.9375 0.9984

Table 6 The regression coefficients for compression-tension. Types of concrete mix

SCC SFSCC 0.5 SFSCC 1.0

Fibre volume fraction, vf (%)

0 0.5 1.0

The fitting parameters

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R.N. Mohamed et al. / Construction and Building Materials 224 (2019) 255–265

Table 7 The regression coefficients for tension-tension. Types of concrete mix

SCC SFSCC 0.5 SFSCC 1.0

2.

3.

4.

5.

Fibre volume fraction, vf (%)

0 0.5 1.0

of 0.05. Meanwhile, under biaxial tension, the strength of SFSCC decreased when the stress ratio exceeded 0.05. The stress ratio of biaxial loading progressively decreased as the strength of SFSCC decreased until the minimum strength values were reached at equal biaxial tension (f1/f2 = 1.0). The effects of the integration of steel fibres into concrete mixtures varied. This variance depended on the stress ratios and test conditions applied. The observed variance was due to the mismatched action between the fibre content and force pressure. Under such a stress-state, the fibres had just experienced relief before they started to resist the load. The confinement effect exercised by the fibres due to their pullout resistance ensures ductility behaviour. When minor principal compressive stress, f2, was introduced, the point at which the stress-strain curve began to deviate from linearity increased due to a confinement effect that resulted from the application of biaxial compression. In the C-T test, the principal failure compressive and tensile strains decreased as the applied tensile load increased due to the associated increase in the absolute value of f2/f1. The failure cracks that formed under uniaxial and biaxial compression were of the splitting and shear failure type, respectively. The specimens under uniaxial and biaxial tension underwent the same types of failure (i.e., the cracks ran perpendicular to the direction of the greater tensile force). The failure criteria of concrete were proposed for each region based on octahedral stress space. These quadratic parabola functions were derived from regression analyses and provide the basis for analysing the performance of SFRC structures under multiaxial stress.

The fitting parameters f

g

h

R2

0.09 0.0679 0.108

3.5000 1.5904 4.8122

50.0000 15.654 70.505

1.0000 0.9426 0.9955

[9]

[10]

[11]

[12]

[13]

[14] [15] [16]

[17]

[18]

[19]

[20] [21]

[22]

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