A Model for the Prediction of the Tensile Strength of FiberReinforced Concrete Members, Before and After Cracking
Abstract
:1. Introduction
2. Materials and Methods
2.1. Model Description
2.2. Volumetric Contribution of Fibers
2.3. Influence of Crack Widening
2.3.1.Geometrical Limitations
2.3.2. Influence of Crossing Angle
3. Results
3.1. Case Study
3.2. Quantification of Volumetric Distribution of Fibers
4. Conclusions
Author Contributions
Conflicts of Interest
Appendix A
A_{c}  area of concrete 
B  pullout resistant force of a fiber 
B_{ud}  design value of maximum pullout force of a fiber, in the case that the fiber reaches the yield stress 
F_{extr,ϕ,perp}  total pullout resistance of fibers forming an angle ϕ perpendicular to crack direction 
N_{ϕ}  number of fibers that cross a crack, forming an angle ϕ perpendicular to crack direction 
d_{f}  diameter of a fiber 
h_{f}  hook height (Figure 1) 
f_{ctm}  tensile strength of concrete 
f_{bd}  ultimate bond stress between fiber and concrete (EC21, cl. 8.4.2) 
f_{sy}  tensile yield stress of a fiber 
f_{α}  the bearing capacity of concrete against compression inside a bent of a fiber, that can be calculated as (EC21, cl. 8.4.2):
$${f}_{\alpha}=\frac{1.5\xb7{f}_{ck}}{1+\frac{2\xb7{d}_{f}}{{a}_{b}}}$$

α_{b}  the distance (perpendicular to the crack plane) between axes of adjacent fibers. In case of uniformly oriented fibers, this can be efficiently approximated by
$${a}_{b}=\sqrt{\frac{{A}_{c}}{\Sigma \left(n\right)}}\text{}\approx \sqrt{\frac{1}{ct}}\approx 0$$
Then
$${f}_{a}\text{}\approx 1.5\xb7\text{}{f}_{ck}$$

α_{ϕ}  modification factorof the volumetric ratio, regarding angle φ 
l_{f}  total length of fiber (refer to Figure 1) 
l_{1}  length of middle part of fiber (refer to Figure 1) 
l_{av}  for one fiber, is the length of the smaller of the two parts of it, as divided by a crack plane; for many fibers is the average of the smaller parts, practically ${l}_{av}=\frac{1}{2}\xb7\left(\frac{{l}_{1}}{2}+{l}_{2d}+{l}_{2v}\right)$(refer to Figure 1) 
n  number of effective fibers that cross area A (function of the volumetric ratio) 
r  effectiveness ratio for fibers [refer to section 3.2] 
μ  friction coefficient between fiber and concrete, considered here as μ = 0.45. 
ρ  volumetric ratio of fibers 
ρ_{act}  active volumetric ratio of fibers 
τ_{fr}  friction due to compression inside a bent fiber, τ_{fr} = μf_{α} 
ϕ  angle formed by a fiber perpendicular to crack direction 
Appendix B
Property  Quantity  Property  Quantity 

Diameter d_{f} (mm)  0.75  Long. dim. of hook h_{f} (mm)  2.50 
Total longitudinal dimension l_{f} (mm)  29.0  Angle of hook θ (°)  26.5 
Central straight part l_{1} (mm)  14.0  Total Length l_{tot} (mm)  30.2 
Long. dimension of hook (mm)  7.50  Area A_{f} (mm^{2})  0.44 
Horizontal part l_{2,h} (mm)  2.49  Volume V_{f} (mm^{3})  13.3 
Diagonal part l_{2,d} (mm)  5.60  Vol. of straight part V_{1} (mm^{3})  6.18 
Horizontal T l_{2,d’} (mm)  5.01  Volumetric ratio  0.50% 
Capacity to tension f_{sy} (MPa)  1150  Modulus of elasticity E_{s} (MPa)  2 × 10^{5} 
Property  Quantity  Property  Quantity 

Length L (mm)  600  Compressive Strength f_{ck} (MPa)  25.0 
Width B (mm)  300  Tensile Strength f_{ctm} (MPa)  2.565 
Height H (mm)  300  Modulus of Elasticity (GPa)  30.5 
References
 Sorensen, C.O.; Berge, E.A.; Saga, P.E.; Østvold, A. Factors affecting the efficiency of fibers in concrete on crack reduction. Open J. Civ. Eng. 2013, 3, 80–85. [Google Scholar] [CrossRef]
 Buratti, N.; Mazzotti, C.; Savoia, M. Postcracking behavior of steel and macrosynthetic fiberreinforced concretes. Constr. Build. Mater. 2011, 25, 2713–2722. [Google Scholar] [CrossRef]
 Mehran, K.; Majid, A. Use of glass and nylon fibers in concrete for controlling early age micro cracking in bridge decks. Constr. Build. Mater. 2016, 125, 800–808. [Google Scholar]
 Boscato, G.; Russo, S. Experimental investigation on repair of RC pavements with SFRC. In Proceedings of the 2nd International Conference on Concrete Repair, Rehabilitation and Retrofitting (ICCRRR), Cape Town, South Africa, 24–26 November 2008; pp. 449–450. [Google Scholar]
 Kurihara, N.; Kunieda, M.; Kamada, T.; Uchida, Y.; Rokugo, K. Tension softening diagrams and evaluation of properties of steel fiber reinforced concrete. Eng.Fract. Mech. 2000, 65, 235–245. [Google Scholar] [CrossRef]
 Holschemacher, K.; Mueller, K.; Ribakov, Y. Effect of steel fibers on mechanical properties of highstrength concrete. Mater. Des. 2010, 31, 2604–2615. [Google Scholar] [CrossRef]
 Yang, J.M.; Min, K.H.; Shin, H.O.; Yoon, Y.S. Effect of steel and synthetic fibers on flexural behavior of high strength concrete beams reinforced with FRP bars. Compos. Part B 2012, 43, 1077–1086. [Google Scholar] [CrossRef]
 Meda, A.; Minelli, F.; Plizzari, G.A. Flexural behavior of RC beams in fiber reinforced concrete. Compos. Part B 2012, 43, 2930–2937. [Google Scholar] [CrossRef]
 Soutsos, M.N.; Lampropoulos, A.P. Flexural performance of fiber reinforced concrete made with steel and synthetic fibers. Constr. Build. Mater. 2012, 36, 704–710. [Google Scholar] [CrossRef]
 Olivito, R.S.; Zuccarello, F.A. An experimental study on the tensile strength of steel fiber reinforced concrete. Compos. Part B 2010, 41, 246–255. [Google Scholar] [CrossRef]
 Choumanidis, D.; Badogiannis, E.; Nomikos, P.; Sofianos, A. The effect of different fibers on the flexural behaviour of concrete exposed to normal and elevated temperatures. Constr. Build. Mater. 2016, 129, 266–277. [Google Scholar] [CrossRef]
 Balaguru, P.; Najm, H. Highperformance fiberreinforced concrete mixture proportions with high fiber volume fractions. ACI Mater. J. 2004, 101, 281–286. [Google Scholar]
 Lee, J.H.; Chob, B.; Choic, E. Flexural capacity of fiber reinforced concrete with a consideration of concrete strength and fiber content. Constr. Build. Mater. 2017, 138, 222–231. [Google Scholar] [CrossRef]
 Deluce, J.R.; Lee, S.C.; Vecchio, F.J. Crack model for steel fiberreinforced concrete members containing conventional reinforcement. ACI Struct. J. 2014, 111, 94–102. [Google Scholar]
 Lee, C.; Kim, H. Orientation factor and number of fibers at failure plane in ringtype steel fiber reinforced concrete. Cem.Concr. Res. 2010, 40, 810–819. [Google Scholar] [CrossRef]
 Alberti, M.G.; Enfedaque, A.; Galvez, J.C. On the prediction of the orientation factor and fiber distribution of steel and macrosynthetic fibers for fiberreinforced concrete. Cem. Concr. Compos. 2017, 77, 29–48. [Google Scholar] [CrossRef]
 Lee, S.C.; Oh, J.H.; Cho, J.Y. Fiber orientation factor on rectangular crosssection in concrete Members. IACSIT Int. J. Eng. Technol. 2015, 7, 470–473. [Google Scholar]
 GeorgiadiStefanidi, K.; Mistakidis, E.; Pantousa, D.; Zygomalas, M. Numerical modeling of the pullout of hooked steel fibres from highstrength cementitious matrix, supplemented by experimental results. Constr. Build. Mater. 2010, 24, 2489–2506. [Google Scholar] [CrossRef]
 Afroughsabet, V.; Biolzi, L.; Ozbakkaloglu, T. Highperformance fiberreinforced concrete: A review. J. Mater. Sci. 2016, 14, 6517–6551. [Google Scholar] [CrossRef]
 Baltay, P.; Gjelsvik, A. Coefficient of friction for steel on concrete at high normal stress. J. Mater. Civ. Eng. 1990, 2, 46–49. [Google Scholar] [CrossRef]
Intervals for ϕ  Number of Fibers N_{ϕ} 

(0°–10°)  2.12 
(10°–20°)  6.30 
(20°–30°)  10.29 
(30°–40°)  13.97 
(40°–50°)  17.22 
(50°–60°)  19.95 
(60°–70°)  22.08 
(70°–80°)  23.53 
(80°–90°)  24.27 
Total  139.74 
Crossing Point  Length of Middle Part l_{1}’ (mm)  Total Length l_{av} (mm)  Total Pullout Force B (kN)  Participation Factor 

0 (at the hook)  0  8.089  0.1646  78.8% 
l1/8  1.75  9.839  0.1758  84.1% 
l1/4  3.5  11.589  0.1869  89.4% 
3l1/8  5.25  13.339  0.1980  94.7% 
l1/2  7  15.089  0.2091  100.0% 
Crossing Point  Bond  Compressive Resistance  Friction  

F_{bd}(kN)  Percentage  F_{α}(kN)  Percentage  T_{fr}(kN)  Percentage  
0  0.0513  31.2%  0.0664  40.3%  0.0469  28.5% 
l_{1}/8  0.0624  35.5%  0.0664  37.8%  0.0469  26.7% 
l_{1}/4  0.0735  39.4%  0.0664  35.5%  0.0469  25.1% 
3l_{1}/8  0.0846  42.8%  0.0664  33.5%  0.0469  23.7% 
l_{1}/2  0.0957  45.8%  0.0664  31.8%  0.0469  22.4% 
Angle ϕ (°)  Maximum Axial Force Per Fiber (kN)  Coefficient Perprendicular to Crack (kN)  Maximum Crack Width before Fiber Extraction (mm),in Accordance to Embedded Length  

0 (hook)  $\frac{{\mathit{l}}_{1}}{8}$  $\frac{{\mathit{l}}_{1}}{4}$  $\frac{3{\mathit{l}}_{1}}{8}$  $\frac{{\mathit{l}}_{1}}{2}$  
5°  0.1946  0.1435  6.00  7.30  8.55  9.90  11.15 
15°  0.1994  0.1582  6.45  7.85  9.25  10.65  12.00 
25°  0.2028  0.1710  6.85  8.35  9.80  11.30  12.75 
35°  0.2046  0.1815  7.20  8.80  10.35  11.90  13.45 
45°  0.2048  0.1892  7.50  9.15  10.75  12.40  14.00 
55°  0.2035  0.1941  7.75  9.45  11.10  12.80  14.45 
65°  0.2006  0.1959  7.95  9.65  11.35  13.10  14.85 
75°  0.1962  0.1946  8.10  9.80  11.55  13.30  15.15 
85°  0.1903  0.1902  8.15  9.90  11.65  13.40  15.15 
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Vougioukas, E.; Papadatou, M. A Model for the Prediction of the Tensile Strength of FiberReinforced Concrete Members, Before and After Cracking. Fibers 2017, 5, 27. https://doi.org/10.3390/fib5030027
Vougioukas E, Papadatou M. A Model for the Prediction of the Tensile Strength of FiberReinforced Concrete Members, Before and After Cracking. Fibers. 2017; 5(3):27. https://doi.org/10.3390/fib5030027
Chicago/Turabian StyleVougioukas, Emmanouil, and Maria Papadatou. 2017. "A Model for the Prediction of the Tensile Strength of FiberReinforced Concrete Members, Before and After Cracking" Fibers 5, no. 3: 27. https://doi.org/10.3390/fib5030027