Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections
Abstract
:1. Introduction
2. Materials Hypothesis
2.1. Non-Cracked Zone
2.2. Cracked Zone
3. Modeling of Crack Propagation
4. Model Response and Experimental Validation
5. Size Effect on Flexural Strength for FRC
6. Practical Expression to Determine the Flexural Strength in FRC
7. Brittle–Ductile Transition in Flexural Failure for FRC Sections
8. Conclusions
- The planar crack assumption can be considered as an alternative to Navier’s hypothesis to model the FRC cracked zone. Using this approach, we avoid using length parameters as lcs to evaluate strains from crack openings, as is commonly carried out in models based on stress–strain laws.
- We propose a brittleness number, βH,FRC, analogous to the one of Hillerborg, as a characterization parameter of FRC structural sections. It is derived from a nondimensional analysis, which includes the beam size and FRC softening characteristics.
- The model fits experimental results very well. Moreover, the model reproduces the asymptotic behavior expected from plastic limit solution for cohesive cracks—very short depths—to the linear elastic solution—overly large depths.
- We offer an expression to calculate the flexural strength of a fiber-reinforced concrete section based on the model results. It depends on the brittleness number, βH,FRC and on the serviceability residual stress, fFts. Its range of validity covers most of practical cases and thus, it can be profitably used for the structural design of FRC sections.
- The model, also, allows studying the ductile–brittle transition in FRC sections. It depends only on two parameters, namely the related tensile strength of the base concrete, ft, and the brittleness number, βH,FRC.
- The planar crack model contributes to a better understanding of the nature of flexural behavior of FRC sections and gives a more physical approach to their failure behavior. In addition, the expressions derived from the model results can be used for structural engineering purposes, constituting a design toolset that avoids complex calculations through finite elements.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
b | beam width |
Ec | longitudinal elastic modulus of concrete |
fFts | serviceability residual stress of concrete |
fFtu | ultimate residual stress of concrete |
fR | flexural strength |
fR* | non-dimensional flexural strength |
fR1 | residual flexural tensile strength corresponding to CMOD = 0.5 mm |
fR3 | residual flexure tensile strength corresponding to CMOD = 2.5 mm |
ft | tensile strength of concrete |
ft* | non-dimensional tensile strength of concrete |
AF,FRC | area under the softening crack law (energy per unit area to open a crack up to the ultimate crack opening) |
h | beam height |
lcs,FRC | structural characteristic length of FRC |
Li | specific internal length |
M | bending moment |
M* | nondimensional bending moment |
Mcr | critical bending moment |
Mmax | maximum bending moment during crack growth |
Mmax* | maximum nondimensional bending moment during crack growth |
w | crack opening |
w* | nondimensional crack opening |
wb | crack mouth opening displacement (opening of the crack at the bottom edge of the section) |
wb* | non-dimensional crack mouth opening displacement |
wu | ultimate crack opening |
yn | neutral axis depth |
z | crack depth |
z0 | crack depth at critical opening |
α | relation between fFtu and fFts |
βH,FRC | brittleness number for FRC |
γn | non-dimensional neutral axis depth |
ε | strain |
εb | strain at bottom part of the beam |
εt | strain at the top part of the beam |
ξ | non-dimensional crack depth |
ξmax | maximum nondimensional crack depth |
σ | stress |
σb | stress at the bottom part of the beam |
σb* | non-dimensional stress at the bottom part of the beam |
σt | stress at the top part of the beam |
σt* | non-dimensional stress at the top part of the beam |
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Reference | Vf (kg/m3) | Pmax (kN) | fR,1 (MPa) | fR,3 (MPa) | b (m) | h (m) | L (m) | Ec (MPa) | wu (mm) | fFts (MPa) | fFtu (MPa) | Af,FRC (N/mm) | βH |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
40 kg/m3 [38] | 40 | 16.00 | 7.13 | 5.69 | 0.10 | 0.20 | 1.20 | 35,728 | 2.50 | 3.21 | 1.42 | 5.79 | 0.0080 |
Malgorzata-0.57H [39] | 45 | 16.29 | 3.61 | 2.22 | 0.15 | 0.15 | 0.50 | 43,281 | 2.50 | 1.63 | 0.39 | 2.52 | 0.0034 |
Malgorzata-0.57CH [39] | 45 | 16.49 | 3.27 | 1.78 | 0.15 | 0.15 | 0.50 | 41,903 | 2.50 | 1.47 | 0.24 | 2.13 | 0.0035 |
Michels. 0.65% [40] | 51 | 31.40 | 7.73 | 6.93 | 0.15 | 0.15 | 0.60 | 30,000 | 2.50 | 3.48 | 1.92 | 6.75 | 0.0062 |
Michels. 0.52% [40] | 41 | 30.60 | 7.47 | 6.67 | 0.15 | 0.15 | 0.60 | 30,000 | 2.50 | 3.36 | 1.84 | 6.50 | 0.0061 |
Zhang (78.4 kg/m3) [41] | 78 | 16.50 | 9.60 | 6.00 | 0.10 | 0.10 | 0.40 | 32,000 | 2.50 | 4.32 | 1.08 | 6.75 | 0.0081 |
Doo-S13 (157 kg/m3) [42] | 157 | 27.90 | 12.15 | 9.00 | 0.10 | 0.10 | 0.30 | 50,876 | 2.50 | 5.47 | 2.07 | 9.42 | 0.0053 |
Doo-S16.3 (157 kg/m3) [42] | 157 | 32.90 | 13.50 | 12.60 | 0.10 | 0.10 | 0.30 | 46,260 | 2.50 | 6.08 | 3.60 | 12.09 | 0.0043 |
Doo-S19.5 (157 kg/m3) [42] | 157 | 37.90 | 14.85 | 15.75 | 0.10 | 0.10 | 0.30 | 46,126 | 2.50 | 6.68 | 4.91 | 14.48 | 0.0031 |
Barros (60 kg/m3) [43] | 60 | 11.50 | 2.20 | 1.90 | 0.15 | 0.15 | 0.45 | 33,366 | 2.50 | 0.99 | 0.51 | 1.88 | 0.0017 |
Barros (45 kg/m3) [43] | 45 | 7.50 | 1.40 | 1.20 | 0.15 | 0.15 | 0.45 | 33,935 | 2.50 | 0.63 | 0.32 | 1.19 | 0.0011 |
Ali (60 kg/m3) [44] | 60 | 28.00 | 7.00 | 5.67 | 0.15 | 0.15 | 0.60 | 34,484 | 2.50 | 3.15 | 1.43 | 5.73 | 0.0060 |
Ali (40 kg/m3) [44] | 40 | 28.00 | 7.33 | 6.33 | 0.15 | 0.15 | 0.60 | 35,808 | 2.50 | 3.30 | 1.70 | 6.25 | 0.0054 |
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Carmona, J.R.; Cortés-Buitrago, R.; Rey-Rey, J.; Ruiz, G. Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections. Materials 2022, 15, 5821. https://doi.org/10.3390/ma15175821
Carmona JR, Cortés-Buitrago R, Rey-Rey J, Ruiz G. Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections. Materials. 2022; 15(17):5821. https://doi.org/10.3390/ma15175821
Chicago/Turabian StyleCarmona, Jacinto R., Raúl Cortés-Buitrago, Juan Rey-Rey, and Gonzalo Ruiz. 2022. "Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections" Materials 15, no. 17: 5821. https://doi.org/10.3390/ma15175821