Abstract
It is well known that the inclusion of steel fibres into a concrete matrix has the potential to improve both the serviceability and strength characteristics of reinforced concrete. As a result of decades of academic interest into the material, design provisions for flexural strength, shear strength and short-term serviceability have been incorporated into international codes of practice. However, available design guidelines typically contain little or no guidance for engineers to confidently predict the long-term behaviour of structures manufactured with steel fibre reinforced concrete (SFRC). This limits the full utilization of the material in design practice. This paper presents a simplified model, suitable for design, that can reliably predict the effects of creep and shrinkage in SFRC members containing conventional reinforcement which are subjected to a sustained in-service flexural load. Predictions of the proposed model are compared to available data in the literature and are shown to correlate well.
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Abbreviations
- A ct :
-
Area of tensile concrete
- A st :
-
Area of longitudinal tensile steel reinforcement
- b :
-
Width of member
- D :
-
Total depth of section
- d :
-
Effective depth
- d f :
-
Diameter of fibre
- d n :
-
Depth of neutral axis
- dn,RC :
-
Depth of neutral axis of a conventionally reinforced section
- d sc :
-
Depth to compressive reinforcement
- E c :
-
Elastic modulus of concrete
- E s :
-
Elastic modulus of steel reinforcement
- f 0.5 :
-
Residual tensile strength calculated at a crack width = 0.5 mm
- f c :
-
Compressive strength of concrete matrix
- f ct :
-
Tensile strength of concrete matrix
- f w :
-
Residual tensile strength of SFRC
- I cr :
-
Cracked 2nd moment of area
- I e :
-
Effective 2nd moment of area
- I g :
-
Gross 2nd moment of area
- L :
-
Span length
- l f :
-
Length of fibre
- M a :
-
Applied external moment
- M cr :
-
Cracking moment
- M cr,sh0 :
-
Cracking moment on specimen subjected to shrinkage
- n :
-
Modular ratio
- s r :
-
Crack spacing
- t :
-
Elapsed time
- α :
-
Creep amplification factor
- α f :
-
Fibre aspect ratio
- Δ :
-
Member deflection
- Δ0:
-
Instantaneous deflection
- Δ cr :
-
Creep induced deflection
- Δ sh :
-
Shrinkage induced deflection
- ε cc :
-
Concrete creep strain
- ε ce :
-
Elastic strain in concrete
- ε sh :
-
Concrete shrinkage strain
- ε shd :
-
Drying shrinkage strain
- ε she :
-
Autogenous shrinkage strain
- γ :
-
Fibre bending angle
- κ :
-
Curvature
- κ 0 :
-
Instantaneous curvature
- κ 0,TS :
-
Tension stiffening curvature offset
- κcr(t):
-
Creep induced strain
- κ L :
-
Curvature at left support
- κ M :
-
Curvature at midspan
- κ R :
-
Curvature at right support
- κsh(t):
-
Shrinkage induced strain
- κ sh,cr :
-
Shrinkage induced strain to cracked concrete
- κ sh,uncr :
-
Shrinkage induced strain to uncracked concrete
- ρ :
-
Tensile longitudinal reinforcing ratio
- ρ′ :
-
Compressive longitudinal reinforcing ratio
- σc :
-
Stress in concrete
- τb :
-
Fibre bond stress
- φ(t, t0):
-
Creep at time t for load applied at t0
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Acknowledgements
This work was supported by an Australian Research Council Discovery Grant (DP 200102114) awarded to the second and third Authors.
Funding
This work was funded by the Australian Research Council Discovery Grant (Grant ID: DP 200102114) awarded to the second and third Authors.
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Appendix 1: Ng et al. [38] model
Appendix 1: Ng et al. [38] model
This Appendix contains additional information that has been provided in the main body of the paper for the sake of brevity.
The model of [38] has been used in Sect. 6 to predict f0.5 in the absence of experimental data. Over a plane of unit area, the stress carried by steel fibres, fw, for a given crack width, w, is taken as:
In Eq. 9, αf is the fibre aspect ratio (lf/df), ρf is the volumetric dosage of steel fibres and Kf is a fibre orientation factor. For Mode I fracture in a 3D domain, Kf is taken as:
In Eq. 10, γ represents the angle between an individual fibre and the loading direction [38] empirically determined γ as:
In Eq. 9, τb is the bond stress for the engaged steel fibres and was empirically approximated as:
f0.5 for a given fibre type (lf, df) and dosage (ρf) may be determined from Eq. 9 by substituting w = 0.5 mm into Eqs. 10 and 11.
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Watts, M.J., Amin, A., Gilbert, R.I. et al. Simplified prediction of the time dependent deflection of SFRC flexural members. Mater Struct 53, 48 (2020). https://doi.org/10.1617/s11527-020-01479-8
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DOI: https://doi.org/10.1617/s11527-020-01479-8