Effect of steel fibres in combination with different reinforcing ratios on the performance of continuous beams

Construction and Building Materials 227 (2019) 116553

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Effect of steel fibres in combination with different reinforcing ratios on the performance of continuous beams Frank Küsel ⇑, Elsabe Kearsley Department of Civil Engineering, University of Pretoria, Pretoria, South Africa

h i g h l i g h t s
Significant moment redistribution occurs before plastic behaviour.
Steel fibres minimise deflections at lower loads.
The shape of the moment curvature relationship relates to moment redistribution.
Fibre effectiveness can increase with increasing reinforcing bar ratios.

a r t i c l e

i n f o

Article history: Received 20 December 2018 Received in revised form 13 July 2019 Accepted 23 July 2019

Keywords: Fibre reinforced concrete Moment redistribution Deflections Continuous beams

a b s t r a c t The ability of statically indeterminate structures to redistribute moments, thus fully utilising the capacity of non-critical sections, leads to improved structural efficiency. The effect of steel fibres on moment redistribution in reinforced concrete beams was investigated to determine whether the addition of fibres affects the ability of beams to redistribute moments. The results of fifteen two-span continuous beams, each containing a unique combination of steel fibres and reinforcing bars, indicated significant moment redistribution before plastic behaviour. An optimum 1.5% fibre content in terms of moment redistribution corresponded with a flatter post-peak moment-curvature relationship. The addition of fibres led to reduced deflections, with fibre effectiveness increasing with increasing reinforcing ratios. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The danger of brittle failure in concrete structures is averted by suitable design with steel reinforcing ensuring ductile behaviour. Steel fibres are used in concrete to prevent brittle failure, particularly in higher strength concretes. The ductility provided by steel bar reinforcing allows possible redistribution of forces in statically indeterminate structures. Moment redistribution results in a change in the distribution of moments in a structure which differ from the moments obtained from an elastic analysis [1–7]. This behaviour allows for the utilisation of the full capacity of noncritical sections, leading to a simplification in reinforcement requirements at critical sections. An incentive for using steel fibres is to reduce the reinforcing ratio, while minimising cracks and deflections. Flexural performance may be improved with the addition of steel fibres, where the flexural stiffness after cracking is improved [8], although the increase in flexural strength may not be as significant [9]. Typical ⇑ Corresponding author. E-mail address: [email protected] (E. Kearsley). https://doi.org/10.1016/j.conbuildmat.2019.07.279 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.

load-deflection curves in statically indeterminate structures reinforced with ductile steel exhibit three behavioural stages. The first stage is mostly linear before cracking, followed by a change in slope as the number of cracks increases. The final stage occurs when the yield point of the steel has been reached [10–13]. Once the yield point at the critical section has been reached, a large increase in deflections and rotations occur. This behaviour, where rotations occur at the plastic hinge whilst maintaining the moment capacity, is what permits additional loads to be applied to the structure and to be redistributed towards non-critical sections, which is the concept of moment redistribution. Redistribution of moments occurs at all limit states [5,6,14] as a result of changes in the stiffness along the length of the beam due to crack formation and variations in the reinforcing layout [15]. As a result, moment redistribution can occur in brittle structures such as GFRP and CFRP concrete beams [11,16,17], where the redistribution of loads is achieved by the difference in flexural stiffness between the hogging and sagging regions created by the different reinforcing ratios. Moment redistribution therefore occurs at the onset of cracking and may be similar for brittle and ductile reinforcing materials. The degree of moment redistribution however

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

remains constant with increasing loads, up to brittle failure, where a secondary increase in moment redistribution is only possible during plastic behaviour in concrete reinforced with ductile materials [6,18]. A moment curvature relationship in which a plateaux is reached after the peak moment is ideal for moment redistribution. The ability of steel fibres to contribute towards moment redistribution therefore depends on the post-peak moment curvature behaviour of the fibres, where a flatter slope, tending towards deflection hardening behaviour, is optimal [4]. Literature reviewed on the effect of steel fibres in combination with reinforcing bars provides seemingly contradictory results, a contradiction which stems from the changes in ductility caused by the addition of steel fibres. The addition of steel fibres has been shown to significantly reduce crack widths and increase the number of cracks [19,20]. This behaviour improves ductility, and hence means that the minimum reinforcing ratio can be reduced. In contrast, the fibres may also lead to reduced ductility through a crack localisation phenomenon, and hence possibly less moment redistribution [21–23]. In this case, an increase in the reinforcing ratio will reduce crack localisation as the behaviour tends to strain hardening [22]. The reduced ductility of lightly reinforced beams with added fibres is evidence of the detrimental effect of fibres if the deflection softening shape caused by the fibres becomes dominant [23]. However, certain fibre contents may lead to an increased number of cracks even if they exhibited deflection softening behaviour [9,24]. Ultimately, the crack bridging ability of steel fibres results in reduced deflections because of the inhibited crack growth. As a result, the neutral axis depth of the beam cross-section is reduced with the addition of fibres, particularly at lower loads [25–27]. The effectiveness of the fibres varies with different reinforcing ratios and deflections [28]. The effect of steel fibres in combination with reinforcing bars may therefore lead to contradictory results, where for instance a deflection softening fibre behaviour may or may not lead to crack localisation, depending on the relative reinforcing bar ratio [23]. The aim of this study was therefore to investigate the effect of different steel fibre and reinforcing bar ratio combinations on moment redistribution, as well as changes in the structural performance. The changes in behaviour were discussed in light of the material properties, thus providing a greater understanding of the expected outcome when adding fibres to high strength concrete (80 MPa) reinforced with steel bars. 2. Experimental programme The experimental programme was set up to investigate the changes in structural behaviour of statically indeterminate reinforced concrete beams caused by steel fibre and reinforcing bar combinations. In addition to the continuous beam tests, material tests were performed to quantify the compressive and tensile properties of the beam constituents. An inverse analysis was used to calculate simplified tensile stress–strain responses based on the results of flexural tests. 2.1. Beam specimens and test setup To investigate the influence of varying steel fibre and steel reinforcing combinations, fifteen 5.0 m beams with a rectangular cross-section were tested. The combination of steel fibres and reinforcing bars for each beam is shown in Fig. 1, with properties of the fibres and reinforcing bars given in Table 1. Nine of the fifteen beams contained combinations of steel fibres and reinforcing bars, with six beams containing only one of three steel fibres contents, or one of three different reinforcing bar contents (1, 2 or 3 reinforc-

Reinforcing Ratio Variable 0.0 %

0.7 %

1.4 %

2.2 %

0,0 % Fibre Content

2

1,0 % 1,5 % 2,0 %

Fig. 1. Beam samples with steel fibre and steel reinforcing bar contents.

Table 1 Properties of the steel fibres and reinforcing bars used. Reinforcing bars*

Steel Fibres Typical strength Diameter Length Hook length Hook depth Aspect ratio (L/d) *

2

>1100 N/mm 0.5 mm 30 mm ± 1.5 mm 1.5–4 mm 2.0 ± 1 mm 55

Diameter Elastic modulus Yield stress Ultimate stress Yield strain Ultimate strain

12 mm 215 GPa 537 MPa 626 MPa 0.022 0.15

Properties of reinforcing bars were experimentally determined.

ing bars). The beams were identified in an XBY format, where X represents the fibre percentage, B stands for beam, and Y represents the reinforcing ratio. As the purpose of the study was to asses moment redistribution, a shallow section was adopted to prevent shear failure. Beam dimensions and reinforcing layout are shown in Fig. 2. For each beam the reinforcing ratios were identical for the hogging and sagging regions. This reinforcement arrangement was chosen to minimise the changes in flexural stiffness throughout the length of the beam, and thus simplify the analysis of moment redistribution. The redistribution of moments was expected to occur from the hogging to the sagging regions due to the elastic moment ratios created by the locations of the load points in the test setup. A sketch of this five point bending configuration is shown in Fig. 3. 2.2. Instrumentation and testing To measure the distribution of loads a total of six load cells were used as shown in Fig. 3, three to measure all support reactions, two measuring the forces at the point loads, and an additional load cell to monitor the total applied load. The height of each support could be adjusted to ensure a correct load distribution over the supports before loading to resemble the forces one would expect in an ideal indeterminate beam setup. Hence, moments induced in the beam as a result of an imperfect test setup, instead of the applied load, were minimised. Two Linear Variable Transistors (LVDTs) were placed at the load points in the sagging regions to measure the vertical deflections relative to the centreline of the beam at the supports. The beams were tested in deflection control at a rate of 0.833 mm/min using a close-loop Materials System Testing (MTS) with a load capacity of 250 kN. 3. Materials Four batches of high performance concrete with fibre contents of 0 kg/m3, 60 kg/m3, 120 kg/m3, and 180 kg/m3 were prepared for the beams. The mix designs of each batch can be seen in Table 2. The volume of sand and superplasticisers had to be varied to ensure adequate workability with an increase in the fibre content.

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

3

Fig. 2. Beam dimensions and reinforcing layout.

Fig. 3. Indeterminate beam test setup.

3.1. Compression behaviour

Table 2 Mix design. Material

RD

0% Fibres (kg/m3)

1.0% Fibres (kg/m3)

1.5% Fibres (kg/m3)

2.0% Fibres (kg/m3)

Cement Dolomite Stone Dolomite Sand Water Superplasticiser Steel Fibres Total

3.14 2.85 2.85 1 1.06 7.9

700 790 808 210 7.0 0 2515

700 790 776 210 7.7 80 2563

700 790 761 210 10.5 120 2591

700 790 744 210 11.2 160 2615

The compressive behaviour was determined from three 150 mm  150 mm cubes for each concrete batch. Testing was done at a loading rate of 0.3 MPa/s. Elastic moduli were obtained from two 150 mm  300 mm cylinders which were loaded to 40% of the characteristic compressive strength before being unloaded. This process was repeated three times per cylinder. Table 3 lists the main compressive properties of each batch, with the water cured samples included as reference. 3.2. Flexural behaviour

To describe the stress-strain behaviour of the beams, three material property tests were conducted on each concrete batch, namely compression tests, elastic modulus tests, and Four Point bending Tests (FPBTs). The large sizes of the 5.0 m beams meant that water curing was not possible. Beams were therefore wrapped in plastic after casting and stored under a thermal blanket to ensure consistency in the concrete strength. The samples used to obtain the material properties were cast from the same batch as that of the continuous beams, with half of the material samples kept in the same environment to reflect the actual properties of the beams, and the other half cured in water at 25 °C as a control. Material tests were conducted when the beams were tested no less than 28 days after casting.

The tensile properties of the concrete were determined from flexural FPBTs on account of the simplicity of the setup. Although indirect tensile tests do not provide tensile stress-strain results, load deflection (P-d) and moment curvatures (M-u) were measured which were valuable when considering the expected structural performance of the fibre reinforced concrete. Vertical deflections were recorded with LVDTs on either side of the beam, and curvatures were measured using Particle Image Velocimetry (PIV). FPBTs were conducted on three air cured 100 mm  100 mm  450 mm beams for each concrete batch in deflection control at a rate of 0.3 mm/min. The P-d and M-u results for all beams are shown in Fig. 4 to indicate the changes in the post-peak behaviour due to the inclusion of

4

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

Table 3 Compressive material properties. Concrete Batch

0.0% Fibres

Cube 1 Stress (MPa) Cube 2 Stress (MPa) Cube 3 Stress (MPa) Average max stress (MPa) Average strain at max stress (e) Cylinder 1 Elastic modulus (GPa) Cylinder 2 Elastic modulus (GPa) Average Elastic modulus (GPa) *

1.0% Fibres

*

1.5% Fibres

*

74.5 (77.7) 76.8 (85.9)* 78.5 (87.1)* 76.8 (83.6)* 0.00323 (0.00327)* 36.3 (44.6)* 36.2 (44.0)* 36.3 (44.3)*

2.0% Fibres *

84.1 (93.9) 81.5 (90.4)* 80.5 (90.5)* 82.0 (91.6)* 0.00412 (0.00384)* 36.4 (41.8)* 37.8 (41.8)* 37.1 (41.8)*

84.1 (101.4)* 77.4 (104.1)* 82.4 81.3 (102.8)* 0.00444 (0.00365)* 37.5 (44.7)* 36.8 (38.1)* 37.2 (41.4)*

85.8 (102.0) 83.0 (103.1)* 86.5 (97.9)* 85.1 (101.0)* 0.00402 (0.00380)* 37.4 (45.5)* 38.2 (45.0)* 37.8 (45.2)*

Water cured samples.

50

1.0 % Fibre 1.5 % Fibre 2.0 % Fibre

2,0

2.0 % Fibres 1,5 % Fibres 1.0 % Fibres 0.0 % Fibres

40

Load (kN)

Moment (kNm)

2,5

1,5 1,0

30 20

10

0,5

0

0,0 0,0

0,2

0,4

0,6

0,8

1,0

Curvature (1/m)

0

1

2

3

4

5

Deflection (mm)

Fig. 4. Average (a) moment curvature and (b) load deflection responses from FPBTs.

the steel fibres. Only the load-deflection behaviour of the beam without fibres is shown since almost no curvature could be measured before the peak load, after which sudden brittle failure occurred. A deflection softening shape was observed for all fibre contents. These load deflection and moment curvature responses were used to back-calculated a simplified tensile stress-strain

response. An increase in the fibre content resulted in higher moment capacities. The descending slope after the peak moment becomes steeper in the beam containing 2.0% fibres, indicating that an optimum fibre content exists near 1.5%, beyond which a further increase in fibres content results in relatively smaller improvements in behaviour.

Fig. 5. Calculated M-u and P-d responses compared against experimental results.

5

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

3.3. Inverse analysis

7

700

(0)

600

6

500

(2)

400 300 200

5

Steel bar 1 Steel bar 2 1.0 % Fibres 1.5 % Fibres 2.0 % Fibres

4 3

(3)

(1)

2

100

1

(4)

0 0,00

0,02

0,04

0,06

SFRC Tensile Stress (MPa)

Reinforcing Bar Tensile Stress (MPa)

A generalised analytical method of inverse analysis, similar to the one proposed by Elsaigh [29] was used. An iterative trial and

0 0,10

0,08

error process is followed by adjusting the assumed tensile stressstrain response until the analytical M-u or P-d results closely match the experimental results. Owing to the high strength of the concrete, a simplified bilinear compressive stress-strain relationship was assumed using the elastic modulus and compressive stress obtained from the material tests. In calculating a theoretical M-u relation, plane sections are assumed to remain plane during bending, and equilibrium is assumed to exist between the internal stresses and externally applied loads. These assumptions allow solutions to be obtained by varying the neutral axis so that equilibrium of the crosssection is achieved at each loading stage. Moment curvature responses were converted to load deflection responses taking deformations caused by moments and shear distortions into account. A comparison of the calculated M-u and P-d responses against the experimentally obtained responses is shown in Fig. 5. For all cases, a close match was obtained, indicating that the simplified stress-strain relationships used were adequate in describing the different behavioural states of the fibre reinforced concrete. The

Strain (ε) Fig. 6. Simplified stress-strain responses for the fibre reinforced concrete.

Table 5 Summary of load and deflection results. Ultimate load (kN)

Table 4 Stress and strain values for all three fibre contents. 1.0% Fibres

rt0 (MPa); et0 rt1 (MPa); et1 rt2 (MPa); et2 rt3 (MPa); et3 rt4 (MPa); et4

6.44 2.80 3.40 1.35 0.00

1.5% Fibres

1.74e4 1.74e4 9.00e3 2.50e2 1.00e1

6.44 3.00 4.20 1.70 0.00

2.0% Fibres

1.74e4 1.74e4 3.00e3 3.00e2 0.83e1

6.44 3.20 5.05 1.35 0.00

1.74e4 1.74e4 2.00e3 3.50e2 0.80e1

100

0.0 0.7 1.4 2.2

30B2.2 Bars Left L 30B2.2 Bars Right R 20B1.4 Bars Left L 20B1.4 Bars Right R 10B0.7 Bar Left L 10B0.7 Bar Right R Max MaxLoad Load

60 40 20

Fibre Content (%)

1.0

1.5

2.0

27.4 50.2 46.9

9.2 32.5 59.8 79.9

14.8 36.1 61.7 79.9

17.2 36.2 60.7 83.6

0.0

1.0

1.5

2.0

31.9 38.6 17.8

1.0 28.6 43.8 37.8

5.9 27.4 35.6 35.9

6.3 32.9 40.6 34.5

0

10

20

30

40

L 31B2.2 Bars Left 31B2.2 Bars Right R L 21B1.4 Bars Left

80 60

21B1.4 Bars Right R 11B0.7 Bar Left L 11B0.7 Bar Right R 01B0.0 Bar Left L 01B0.0 Bar Right R Max MaxLoad Load

40 20

0

0

50

0

Deflection (mm)

10

20

30

40

50

Deflection (mm)

100

100

60

R 21.5B1.4 Bars Right 11.5B0.7 Bar LeftL

40

11.5B0.7 Bar Right R 01.5B0.0 Bar LeftL

20

R 01.5B0.0 Bar Right Max MaxLoad Load

0 0

10

20

30

40

Deflection (mm)

50

32B2.2 Bars Left L 32B2.2 Bars Right R 22B1.4 Bars Left L 22B1.4 Bars Right R 12B0.7 Bar Left L 12B0.7 Bar Right R 02B0.0 Bars Left L 02B0.0 Bars Right R Max MaxLoad Load

80

Load (kN)

L 31.5B2.2 Bars Left R 31.5B2.2 Bars Right L 21.5B1.4 Bars Left

80

Load (kN)

Fibre Content (%) 0.0

100

80

Load (kN)

Reinforcing ratio (%)

Load (kN)

Parameter

Deflections at ultimate load (mm)

60 40 20 0 0

10

20

30

40

Deflection (mm) Fig. 7. Load-deflection responses of the beams.

50

6

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

output stress-strain responses from the inverse analysis are shown in Fig. 6, with the stress and strain values listed in Table 4 in order of increasing strains. The stress-strain results for the reinforcing bars are plotted on the same strain axis to highlight the difference in tensile behaviour for the two reinforcing materials. The stressstrain shapes indicate a high cracking stress, followed by a sudden drop when the concrete cracks. Further strains result in a residual stress of varying magnitude depending on the amount of steel fibres used. All fibre contents exhibited a strain softening shape, clearly indicating the residual tensile strength provided to the concrete due to the crack bridging ability of the steel fibres.

Deflection/Load (mm/kN)

1,4 No Fibres 1,2

1.0% Fibres 1.5% Fibres

1,0

2.0% Fibres 0,8 0,6 0,4 0,2

4. Indeterminate beam test results

0,0 3 2.2

1 2 0.7 1.4 Reinforcing Ratio

0

To study the effects of different combinations of fibres and reinforcing bars on the performance of the structure, results will be discussed in three parts: loads and deflections, moment redistribution, and energy related results.

Fig. 8. Deflections at the ultimate load.

4.1. Loads and deflections

0,25 0,20

Steel yields

60

0,15

40

0,10

20

0,05

The load-deflection responses of all beams are shown in Fig. 7, with the maximum loads and deflections at the peak loads listed in Table 5. Deflections normalised with respect to the ultimate loads are shown in Fig. 8 to illustrate the change in loaddeflection behaviour caused by a varying fibre and steel reinforcing ratio. In general, three stages can be identified in the shape of the load-deflection responses: (1) linear elastic before cracking, (2) a change in slope when the concrete cracks, and (3) a change in slope when the steel yields at the critical section. Each of these stages can be clearly determined from the measured horizontal tensile strains across each critical section as shown in Fig. 9. Beam 0B2.2 failed at a lower load than expected due to concrete compaction issues. Similar issues, although very minor occurred in beam 0B0.7 and 0B1.4. These beams will therefore be excluded in the discussion of certain results. Average increases in strength when adding a reinforcing bar for beams with 1.0%, 1.5% and 2.0% fibres were 22.3 kN, 21.7 kN and 22.1 kN respectively. These values were slightly lower than the 27.4 kN strength of beam 0B0.7 representing the strength of a

0,00

0 0

10

20 30 Deflection (mm)

40

50

Load

Tension Sag (R)

Tension Sag (L)

Tensions Hog

Fig. 9. Load-deflection and strain results of beam 2B2.2.

Moment (kNm)

4

2 0 -2

1B1

-4 0

1

2

3

4

Position (m) 20

Moment (kNm)

Load (kN)

80

Microstrain (με)

100

10 0

M Elastic Max Load 75 % Max Load 50 % Max Load

-10

2B3

-20 0

1

2

3

Position (m) Fig. 10. Examples of bending moment diagrams.

4

7

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

beam with only one reinforcing bar. The addition of fibres therefore did not lead to any significant changes in the ultimate load capacities. The addition of steel fibres in combination with reinforcing bars did however reduce the deflections at lower loads, which can be seen from the increase in the slope of the load deflection graph during the second stage of deformation. For instance, the deflection in beams 1B0.7, 1.5B0.7 and 2B0.7 at a load of 15 kN (roughly 50% of the ultimate load) were reduced by 1.6, 2.2, and 2.8 times in comparison to the beam containing only one reinforcing bar (0B0.7). Even in beams with a reinforcing ratio of 2.2%, deflections were reduced by 1.3, 1.6 and 1.8 times, at a load of 30 kN (about 37% of the ultimate load), in beams 1B2.2, 1.5B2.2 and 2B2.2 when compared to the beam without any fibres (0B2.2). The loads of 15 kN and 30 kN were chosen as examples as they corresponded to deflections of about 9 mm in beams 0B0.7 and 0B2.2 respectively, which represents the span/250 deflection limit

recommended by EC2 [30] for the appearance and efficiency of a structure. 4.2. Moment redistribution According to EN 1992-1-1 [30], ‘‘moment redistribution is the ratio of the redistributed moment to the elastic bending moment”. The moment redistribution of each beam can therefore be calculated using Eq. (1):

K MR ¼

16

10 8 6 4

8

Elastic Sagging M

4 2 0

60

Sagging M Right

6

0

40

Hogging M Middle

10

2

20

80 100

Elastic Hogging M 0

20

Load (kN)

6 4 2

12 10 8 6 4 2

40

60

0

80 100

20

16 14 12 10 8 6 4 2 0

40

60

16 14 12 10 8 6 4 2 0

1.5B0.7

40

60

80 100

Moment (kNm)

Moment (kNm)

20

0

20

Load (kN) 16

40

60

16 14 12 10 8 6 4 2 0

80 100

0

Moment (kNm)

Moment (kNm)

6 4 2 0

8 6 4 2 0

0

20

40

60

Load (kN)

80 100

0

20

40

60

20

40

60

Load (kN)

16 14 12 10 8 6 4 2 0

80 100

40

60

80 100

1.5B2.2

0

80 100

20

40

60

80 100

Load (kN) 16

2B2.2

14

12 10 8 6 4 2

12 10 8 6 4 2

0 0

20

Load (kN)

2B1.4

14

10

8

80 100

16

12

10

60

8 6 4 2 0

Load (kN)

2B0.7

14

12

40

1.5B1.4

Load (kN) 16

2B0

14

20

12 10

Load (kN)

Moment (kNm)

1.5B0

0

0

80 100

Load (kN)

Moment (kNm)

20

Load (kN)

Moment (kNm)

8 6 4 2 0

Moment (kNm)

0

1B2.2

14

12 10

0

0

16

Moment (kNm)

8

80 100

1B1.4

14

Moment (kNm)

Moment (kNm)

Moment (kNm)

10

60

16

1B0.7

14

12

40

Load (kN)

16

1B0

14

Sagging M Left

12

Moment (kNm)

12

0

0B1.4

14

0B0.7

14

16

ð1Þ

where Mel is the theoretical elastic moment, and Mexp is the experimental moment calculated from the reaction forces of the load cells. The redistribution of moments is expressed visually in terms of the bending moment diagrams shown in Fig. 10. These diagrams

16

Moment (kNm)

Mel  Mexp Mel

0 0

20

40

60

80 100

Load (kN)

Fig. 11. Hogging and sagging moment evolutions.

0

20

40 60 80 100

Load (kN)

8

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

indicate the change in the moment ratios as the load increases. The difference between the ultimate bending moment diagram and the elastic bending moment diagram indicates the degree to which moment redistribution has occurred between the hogging and sagging regions. The moment distribution in beam 1B0.7 is in contrast to the common expectation where a larger experimental moment is expected at both critical sagging regions in comparison to the elastic moment at the maximum load. In this case, the ductility introduced into the beam by the steel fibres was insufficient to form a plastic hinge in which the moment capacity was maintained upon further rotation. Therefore, as soon as the first plastic hinge formed in beam 1B0.7, almost no further increase in external load was possible due to the reduction in moment capacity upon further rotation. The critical moments in beam 1B0.7 were therefore not reached in the left sagging and central hogging regions since the first hinge occurred in the right sagging region. To fully describe the moment redistribution process, the moment evolutions of all beams in terms of the total applied load were compared against the elastic bending moments in Fig. 11. The

hogging moments were expected to increase at a faster rate than the sagging moments until moment redistribution occurred, whereafter the rate of increase in hogging moments should decrease causing the sagging moments to increase at a faster rate. This behaviour was observed in almost all beams. Beam 1B0 was an exception to this behaviour, since the fibres did not allow a hinge to form with sufficient rotational capacity owing to the short length of the fibres. As shown by the load deflection results of beams 1.5B0 and 2B0 in Fig. 7, a small amount of redistribution was possible in beams with 1.5% and 2.0% as there was small increase in load capacity after a crack had formed at the central support. For beams with reinforcing bars, there was a much clearer indication of this change in moment ratios when the steel yielded. Generally, the greatest redistribution of moments occurred during plastic behaviour after the steel yielded. However, since the redistribution of moments is based on the ratio of the ultimate moment to the elastic moments, one cannot ignore the redistribution of moments during elastic behaviour. For instance, an ideal case of the moment evolution similar to what one could expect

Sagging Moment Left

Moment Redistribution (%)

Hogging Moment Load (kN)

Sagging Moment Right

30

30

30

20

20

20

10

10

10

0

0

0

-10

-10

-10

-20

-20

1B0.7

-30

-20

1B1.4

-30 0

10

20

30

40

0

20

40

60

80

30

30

30

20

20

20

10

10

0

0

-10

-10

-20

-20

1.5B0.7

-30

10 0 -10 -20 -30

1.5B1.4

-30 0

10

20

30

40

1B2.2

-30

1.5B2.2

-40 0

20

40

60

80

0

20

30 20 10 0 -10 -20

2B0.7

2B1.4

2B2.2

-30 0

10

20

30

40

Fig. 12. Examples of moment redistribution for each critical section.

40

60

80

100

9

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

reinforcing bars stabilised where an increase in moment was possible. Once the moment was reached that caused the steel to yield, a second sudden change in the neutral axis is observed, which represents the point at which a hinge is formed, and rotations occur to allow for moment redistribution to take place. A third increase in

Fig. 13. Calculated moment curvature response.

60

Neutral Axis Depth (mm)

from the bilinear design curves is shown by the results of beam 2B1.4. The moment curves followed the elastic moment curves until the concrete cracked at about 2 kNm, followed by a slight deviation from the elastic moments as the concrete continued to crack, with some minor redistribution of moments, and finally a clear change in the moment ratios when the steel yields. In contrast, the moment evolutions of 1.5B2.2 showed much greater deviations from the elastic moments before the steel yielded. The point at which plastic moment redistribution begins is therefore not as clear, however the ultimate moment in beam 1.5B2.2 was significantly less than the elastic moment relative to the results shown for beam 2B1.4. Moment redistribution can be plotted against the applied load, for which the results of all beams containing a combination of both fibres and bars are shown in Fig. 12. Beams with only fibres are excluded as the fibres alone were not capable of handling large rotations. It must be noted that although Eq. (1) was used to calculate the percentage moment redistribution, the sign convention was changed so that a redistribution of moments away from the hogging region was visually represented by negative moment redistribution. In the same way positive moment redistribution values indicate increasing moments in the sagging regions so that the results can be interpreted easily following the moment-load evolutions shown in Fig. 11. Ultimately, the aim of the results shown in Fig. 12 was to highlight the moment redistribution evolution throughout loading. It can be observed that moment redistribution already occurred at low loads. This behaviour can be attributed to the change in flexural stiffness throughout the beam. After cracking, the neutral axis stabilised, and hence moment redistribution remained fairly constant until another sudden change in the neutral axis depth occurred when the steel yielded. Using the material properties of the steel fibre reinforced concrete and reinforcing bars, the neutral axis depth at the ultimate moment can be calculated. Since the stress-strain relationships of both the reinforcing bars and the SFRC were determined separately, an assumption was made that the tensile strengths of both reinforcing materials could be added. This assumption led to the calculated M-u response shown in Fig. 13 with key stresses and strains indicated. The shape of the M-u curves for beams with only fibres are similar to the curves obtained from the FPBTs. The curves of the reinforcing bars reflect the strain hardening properties of the steel. The assumption stated previously resulted in combined fibre and reinforcing M-u responses in which the fibres resulted in not only a significant increase in the moment capacity, but also a change in the slope of the M-u response after the peak moment. Depending on the relative strength and efficiency of the reinforcing materials, a balance is reached between the respective behavioural characteristics. If moment redistribution is required, the ideal behaviour would be a bilinear curve where a plateaux is reached after the peak moment. Theoretically, the fibres cause this in beams with higher reinforcing contents. For low reinforcing contents the addition of fibres may lead to a reduction in possible moment redistribution due to the softening behaviour. However, in reality the contribution of fibres in beams containing reinforcing bars will change due to the change in the number and spacing of cracks caused by the strain hardening property of the reinforcing bars. The evolution of the neutral axis depth of the critical section as shown in Fig. 14 was calculated using the same material properties and assumptions made in the inverse analysis method. Once the concrete cracked, the neutral axis depth suddenly changed, correlating with the change in slope of the load deflection and moment evolution relationships discussed previously. In beams with only fibres, there is not much strength gain as the neutral axis depth is reduced. In contrast, the neutral axis depth of beams with only

50 40 30 20 10 0 0

5

10

15

Moment (kNm)

Fig. 14. Calculated neutral axis depths.

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F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

moments is shown in beams with only reinforcing bars when the strains resulting in strain hardening are reached. Adding fibres to the beams containing reinforcing bars resulted in an increase in moment capacity, with a reduction in the neutral axis depth with more fibres added. Additionally, the behaviour after cracking is improved where the fibres limit the change in neutral axis depth due to their crack bridging ability. The percentage moment redistribution of the beams as calculated using Eq. (1), is shown in Fig. 15. An optimum fibre content of 1.5% was observed, which correlates with the optimum fibre content of 1.5% discussed in the results of the FPBTs. Another general trend of increasing moment redistribution was observed with an increase in the reinforcing ratio. The exception to this trend was beam 2B2.2, which may represent the point at which the beam becomes over reinforced and hence tend towards compressive failure and thereby limit the moment redistribution potential. Beams 1B0, 1.5B0 and 2B0 showed much less moment redistribution due to the short length of the fibres. The strain softening behaviour of the fibres did not allow the formation of multiple cracks, and coupled with their short lengths, resulted in failure of the beam occurring once single cracks formed at the centre support and in one span. In general, the addition of fibres led to additional moment redistribution when compared to beam 0B0.7. Exceptions to this trend were observed in beams 1B0.7 and 1B1.4. For the beams with both fibres and reinforcing bars, beam 1.5B2.2 showed

the highest amount of moment redistribution, and beam 1B0.7 showed the least. The measured degree of moment redistribution, expressed in terms of a neutral axis depth (x) over effective depth ratio (d) compared against the limits suggested by EC2 [30], can be seen in Fig. 16. Darker markers indicate an increase in the reinforcing ratio. As expected, the x/d ratio increased as the steel percentage increased. Beam 1B0 displayed less moment redistribution capacity compared to the code. The small x/d ratio was the result of the high concrete compressive strength in comparison to the low tensile resistance provided by the low reinforcement ratio. The code was conservative in all beams containing a combination of fibres and reinforcing bars apart from beams 1B0.7 and 1.5B1.4. The effects of over-reinforcing can be observed in the reduced moment redistribution capacity of beam 2B2.2.

300 250

Energy (J)

10

200 150 100

29,5 25,6 20,0

18,3

50

Moment Redistribution (%)

24,8 18,3

0

20,3

2

10,1

14,4

2,1

2,00 1,50 1,00 0,00 Steel Fibre Content (%)

3,7

1.4

0.7

0

6 Deflection (mm) 0B3 2.2 0B10.7 1.5B0

3,4

7,9

2.2

4

8

10

0B2 1.4 2B0 1B0

Fig. 17. Energy absorption vs deflection for beams with only fibres and reinforcing bars.

Reinforcing Ratio

300

Fig. 15. Moment redistribution.

250 200

30

Energy (J)

Moment redistribution (%)

35

25 20

150 100

15 10

50

5

0

0 0,0

0,1

0,2

0,3

0,4

2

4

6

8

10

Deflection (mm)

x/d 1.0% Fibres

1.5% Fibres

2B32.2

2B2 1.4

2B10.7

2.0% Fibres

EC2

1.5B32.2

1.5B21.4

1.5B10.7

1B32.2

1B21.4

1B10.7

No bars

1B0

Fig. 16. Moment redistribution compared against code predictions.

Fig. 18. Energy absorption vs deflections for beams with fibres and reinforcing bars.

F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

The degree to which the fibres alter the behaviour of beams containing reinforcing bars depends on the efficiency of the fibres. If the fibres are unable to contribute fully, due to for instance a less optimal fibre distribution and orientation caused by the presence of the reinforcing bars or concrete cover less than the length of the fibre, the optimum fibre content will differ for each reinforcing content since the tensile material properties can no longer simply be added to predict the combined behaviour. Changes in the number of cracks and plastic hinge lengths will affect the ability of the fibres to contribute towards the strength of the structure. One can look at the energy absorbed by the beams throughout the loading stages as a method of describing the combined effect of all factors affecting the performance of the structure.

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4.3. Energy The energy absorbed for all beams were calculated and the results of beams containing only fibres or only reinforcing bars are shown in Fig. 17. The energy was calculated as the area under the load-deflection curves, where the deflections used were the average of the two spans. Beams 1.5B0 and 2B0 absorbed more energy than the beam with a reinforcing ratio of 2.2% up to about 5.2 mm and 6.0 mm respectively. Thereafter, the energy absorption capacity of the fibres was limited by their short length, hence the peak load capacities were reached at much lower deflections in beams with only fibres. Beam 1B0 was unable to absorb much energy since the softening behaviour of concrete reinforced with

Fig. 19. Energy absorptions for experimental results and the sum of fibres and reinforcing bars.

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F. Küsel, E. Kearsley / Construction and Building Materials 227 (2019) 116553

1.0% fibres caused the beam to fail soon after the first crack had formed at the centre support. As the deflections increased beyond 6 mm, the reinforcing bars absorbed energy at an increasing rate, a behaviour opposite to that of the fibres. The energy deflection curves for all beams containing a combination of fibres and reinforcing bars are shown in Fig. 18. As expected, an increase in the reinforcing ratio resulted in more energy being absorbed. The effect of different fibre contents was most evident in beams with a reinforcing ratio of 0.7%, as the relatively higher strength of a reinforcing ratio of 1.4% and 2.2% overshadowed the contribution of the fibres. The results of these beams can be compared to the sum of the energy results of beams with only fibres or only reinforcing bars. The energy deflection curves for beams with a combination of fibres and reinforcing bars were compared to the sum of the energies from the beams with only fibres and only reinforcing bars in Fig. 19. If the strength of the fibres and reinforcing bars were fully utilised, the sum of the energies would be comparable to the actual results. In general, the sum of energies was greater than the actual energy absorbed at lower deflections. The exception was for beams containing 1.0% fibres. This can be explained by the fact that only one crack had formed at the centre support and one span in beam 1B0 before failure. The energy absorption of the fibres was therefore limited since the fibres only bridged across two major cracks. However, when combined with a reinforcing bar, the strain hardening behaviour of the reinforcing led to the formation of multiple smaller cracks, which allowed the fibres to bridge those additional cracks and absorb more energy. Hence, the actual energy absorbed for beams with 1% fibre and two or three reinforcing bars actually increased beyond the sum of the energies since the increase in reinforcing ratio led to the formation of more smaller cracks. The effect of fibre content was more pronounced when combined with a reinforcing ratio of 0.7%, since the tensile resistance was more dependent on the fibres due to the small number of cracks. With an increase in the reinforcing ratio, a change in the fibre content had a lesser effect, although the overall efficiency of the fibres improved, due to the increased number of cracks. This behaviour was confirmed by the average crack spacing in beams with different reinforcing ratios. The fibre content was found to have no noticeable effect on the crack spacing, leading to an average crack spacing of 96 mm for a reinforcing ratio of 0.7%, 67 mm for a reinforcing ratio of 1.4%, and 43 mm for a 2.2% reinforcing ratio, regardless of fibre content. The effect of adding steel fibres to concrete reinforced with steel bars was to reduce the deflections at lower loads. The energy related results indicated that although the fibres did improve the initial energy absorption at lower deflections, they become more effective even at higher deflections with more reinforcing bars due to the increased number of cracks. However, this behaviour would have changed if the strain softening characteristic of the fibres had been dominant, in which case the addition of the fibres would have led to crack localisation, and hence a possible reduction in the moment redistribution capacity. The improved compressive ductility as a result of the residual strength of fibres should be considered since the delaying of concrete crushing would also allow greater moment redistribution.

5. Conclusions The aim of this research was to investigate the effects of different percentages and combinations of steel fibres and reinforcing bars on moment redistribution and general changes in the performance of a statically indeterminate structure. Standard material tests were conducted to relate the material properties to the mechanical behaviour. The experimental results indicate that:


The addition of steel fibres did not lead to significant increases in load bearing capacity but caused a reduction in deflections at lower loads.
Significant moment redistribution occurred during elastic behaviour before any plastic moment redistribution occurred.
An optimum fibre content of 1.5% was found for moment redistribution, which corresponded to the shape of the moment curvature relationship of the FPBTs in which a balance was reached between an increased moment capacity and a less steep postpeak moment-curvature slope.
The effect of fibre content reduced with an increase in reinforcing ratio, however fibres became more effective in beams with more reinforcing bars due to the increase in the number of cracks. Ultimately, a balance between the strain softening and strain hardening behaviour of the fibres and reinforcing must be reached to ensure moment redistribution capacities are not reduced through crack localisation caused by the addition of deflection softening fibres. For these tests, the deflection hardening behaviour of the reinforcing bars was dominant, and hence the fibres acted as additional tensile resistance across the cracks instead of completely changing the crack distribution. Declaration of Competing Interest None. Acknowledgements The authors would like to acknowledge The Concrete Institute which funded this project. References [1] D.A. Nethercot, T.Q. Li, B.S. Choo, Required rotations and moment redistribution for composite frames and continuous beams, J. Constr. Steel Res. 35 (1995) 121–163. [2] D.J. Oehlers, M. Haskett, M.S. Mohamed Ali, M.C. Griffith, Moment redistribution in reinforced concrete beams, Struct. Build. 163 (3) (2010) 165–176. [3] D.J. Oehlers, G. Ju, I.S.T. Liu, R. Seracino, Moment redistribution in continuous plated RC flexural members. Part 1: neutral axis depth approach and tests, Eng. Struct. 26 (14) (2004) 2197–2207. [4] D.J. Oehlers, I.S.T. Liu, G. Ju, R. Seracino, Moment redistribution in continuous plated RC flexural members. Part 2: flexural rigidity approach, Eng. Struct. 26 (14) (2004) 2209–2218. [5] R.N.F. do Carmo, S.M.R. Lopes, Ductility and linear analysis with moment redistribution in reinforced high-strength concrete beams, Can. J. Civ. Eng. 32 (1) (2005) 194–203. [6] P. Visintin, M.S. Mohamad Ali, T. Xie, A.B. Sturm, Experimental investigation of moment redistribution in ultra-high performance fibre reinforced concrete beams, Constr. Build. Mater. 166 (2018) 433–444. [7] T. Lou, S.M.R. Lopes, A.V. Lopes, Composites : Part B Neutral axis depth and moment redistribution in FRP and steel reinforced concrete continuous beams, Compos. Part B 70 (2015) 44–52. [8] F.P. Ackermann, J. Schnell, Steel fibre reinforced continuous composite slabs, Compos. Constr. Steel Concr. VI (2011) 125–137. [9] H. Aoude, M. Belghiti, W.D. Cook, D. Mitchell, Response of steel fiber-reinforced concrete beams with and without stirrups, ACI Struct. J. 109 (3) (2012) 359– 367. [10] C.H. Lin, Y.M. Chien, Effect of section ductility on moment redistribution of continuous concrete beams, J. Chin. Inst. Eng. 23 (2) (2000) 131–141. [11] R.J. Gravina, S.T. Smith, Flexural behaviour of indeterminate concrete beams reinforced with FRP bars, Eng. Struct. 30 (2008) 2370–2380. [12] M.A. Aiello, L. Ombres, Moment redistribution in continuous fiber-reinforced polymer-strengthened reinforced concrete beams, ACI Struct. J. 108 (16) (2011) 158–166. [13] A.F. Ashour, M.N. Habeeb, Continuous concrete beams reinforced with CFRP bars, Struct. Build. 161 (6) (2008) 349–357. [14] P. Schumacher, Rotation Capacity of Self-Compacting Steel Fiber Reinforced Concrete PhD Thesis, Darmstadt University of Technology, Germany, 2006. [15] R.H. Scott, R.T. Whittle, Moment redistribution effects in beams, Mag. Concr. Res. 57 (1) (2008) 9–20.

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