# Influence of Fiber Content on Shear Capacity of Steel Fiber-Reinforced Concrete Beams

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## Abstract

**:**

## 1. Introduction

## 2. Existing Models for the Shear and Flexural Capacity of Steel Fiber-Reinforced Concrete (SFRC)

#### 2.1. Ultimate Shear Capacity

_{f}= fiber volume fraction. D

_{f}= fiber bond factor.

_{f}) accounts for the geometry and bond characteristics of the fibers. For steel fibers, it has a value of 1.00 for hooked fibers, 0.75 for crimped fibers and 0.50 for straight fibers as recommended by Narayanan and Darwish [30]; another method to calculate the fiber bond factor is suggested by [31,32] and it is equal to the ratio between the mean fiber–matrix shear stress and the strength in direct tension of the material. This approach is useful when fibers of materials different than steel are used.

_{a}), and the aggregate size (d

_{a}), which is considered in the size effect term (ψ). Yakoub [34] provides two different equations to predict the shear capacity of slender SFRC beams (a/d > 2.5). The first equation (Equation (12)) is a modification to include the effect of steel fibers of the shear capacity proposed by Bažant and Kim [35] for normal-strength reinforced concrete. The expression takes into account the size of aggregates (d

_{a}), the concrete compressive strength (f

_{c}), shear span to depth ratio (a/d), and longitudinal reinforcement ratio (ρ). The second equation by Yakoub [34], Equation (13) is an extension of the expression for the shear capacity of the Canadian Code CSA A23.3-04 [36] to include the contribution of the steel fibers. This expression is a function of the strain at mid-depth of the beam (ε

_{x}) and crack spacing (s

_{x}) as a function of the aggregate size (d

_{a}). Equation (13) does not consider arching action.

_{f}) is used to find the force resultant of the fiber contribution. The result of these procedures is that the ultimate shear strength is calculated by the summation of Equations (9) and (10). Similarly, Mansur et al. [39] conducted an experimental program to provide an expression to predict the shear capacity of SFRC by adding the contribution of fibers (V

_{sf}) to the concrete contribution (V

_{c}) as calculated in Equation (15). Both Dinh et al. [37] and Mansur et al. [39] use similar expressions for V

_{sf}and include similar parameters such as the tensile strength of concrete (f

_{t}), the geometry of the beam, and the diagonal crack angle (taken as 30 degrees by [37] and 45 degrees by [39]). On the other hand, the expressions for the concrete contribution are based on different assumptions. Dinh et al. [37] consider a uniform shear stress over the depth of the compression zone, whereas Mansur et al. [39] consider the ratio of external shear to moment according to the recommendation of the ACI-ASCE Committee 426 [40].

_{sp}, (2) dowel action provided by the longitudinal reinforcement and taking into account the influence of the shear span to depth ratio, and (3) the fiber pullout stresses along the inclined cracks, v

_{b}. Arching action is taken into account by using the factor e, but small differences exist between Equations (17) and (19), and the effect of arching action is not considered in Equation (20).

_{c}), fiber factor (F), longitudinal reinforcement ratio (ρ), and shear span to depth ratio (a/d). Based on testing high strength (f

_{c}about 93 MPa) SFRC beams with variable hooked-end steel fiber (aspect ratio of 75) content, longitudinal reinforcement ratio, and shear span to depth ratio (a/d), Ashour et al. [42] developed two expressions: (1) Equation (21), an extension of Zsuty’s equation [43] to include the contribution of the fibers through the fiber factor F, and (2) Equation (22), an extension of the ACI 318-89 [44] shear equation to include the contribution of the fibers, as well as the effect of the shear span to depth ratio and the longitudinal reinforcement ratio. The factor 0.7 accounts for the action of high strength concrete. Khuntia et al. [45] developed Equation (23) based on 10 different experimental programs in which the main variables were concrete compressive strength (f

_{c}), shear span to depth ratio (a/d), fiber factor (F), fiber content (V

_{f}), and longitudinal reinforcement ratio (ρ). The expression sums the concrete contribution from ACI 318-95 [46] and the contribution of the fibers, assuming a diagonal crack of 45 degrees. The arching action that is developed when a/d is less than 2.5 is taken into account in the factor α.

_{c}), effective depth (d), shear span to depth ratio (a/d), longitudinal reinforcement ratio (ρ), and fiber factor (F). The model resulted in Equation (24) were the coefficients c

_{0}, c

_{1}, c

_{2}, and c

_{3}are constants provided by the formulation of the GEP model.

#### 2.2. Sectional Shear at Inclined Cracking Load

#### 2.3. Flexural Capacity

## 3. Materials and Methods

#### 3.1. Materials

#### 3.2. Test Setup and Instrumentation

## 4. Results

#### 4.1. Experimental Results

_{cr}), the load that was applied at the moment of failure (P

_{u}), the maximum sectional shear force calculated by the sum of the sectional shear caused by the applied load and the self-weight of the beam (which can be considered negligible) (V

_{u}), the normalized shear stress, the deflection at failure (δ

_{u}), and the failure mode that occurred for each test.

#### 4.2. Comparison to Predicted Shear Capacities

#### 4.3. Analysis of Influence of Fiber Content on Shear Capacity

^{2}value of 0.1363, which show that the influence of fibers is not very representative as shown in Figure 12. The presence of fibers has an influence on this parameter which is different from the results of our experimental program. However, our experiments follow this trend when the inclined cracking load is considered. These observations further underline the need for a better understanding of the mechanics of the different shear-carrying contributions in SFRC, so that recommendations for fiber contents can be based on sound mechanical concepts.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

a/d | shear span to depth ratio |

b_{w} | width of the beam |

c | height of the compression zone |

d | effective depth |

d_{a} | maximum aggregate size |

d_{v} | internal lever arm |

f_{c} | concrete compressive strength |

f_{c}′ | design concrete compressive strength |

f_{cuf} | cube compressive strength of steel fiber-reinforced concrete (SFRC) |

f_{ctf} | peak tensile stress of SFRC |

f_{r} | residual strength of SFRC |

f_{sp} | split tensile strength of SFRC |

f_{y} | longitudinal steel yield strength |

h | height of the cross section |

s_{xe} | equivalent crack spacing factor |

s_{x} | crack spacing parameter |

v_{b} | fiber contribution to shear strength |

v_{cr} | inclined shear capacity |

v_{u} | ultimate shear capacity |

w | crack width |

A_{s} | area of longitudinal steel reinforcement |

C_{c} | resultant of concrete under compression |

D | diameter of the fiber |

D_{f} | fiber bond factor = 1.00 for hooked fibers, 0.75 for crimped fibers, 0.5 for straight fibers |

E_{ct} | elastic modulus of SFRC in tension |

L | length of the fiber |

M | bending moment |

M_{n} | moment capacity of the cross section |

P | applied load |

P_{cr} | inclined cracking load |

P_{u} | ultimate load |

T_{f} | resultant of fibers under tension |

T_{s} | resultant of steel under tension |

V_{c} | shear force carried by the concrete |

V_{cr} | inclined cracking force |

V_{f} | fiber volume fraction |

V_{sf} | shear force carried by the steel fibers |

V_{u} | ultimate shear force |

α | arching action factor for Khuntia et al. [45] |

β | factor that accounts for the strain at mid-depth and aggregate size for Yakoub |

β_{1} | Whitney’s stress block coefficient |

δ_{u} | deflection at ultimate load |

ε_{o85} | compressive strain measured at 0.85f_{c} after peak |

ε_{s} | strain in longitudinal steel reinforcement |

ε_{x} | strain at mid-height of the cross section |

ε_{cu} | concrete ultimate strain |

η | factor that accounts for the effect of fiber in moment capacity |

λ | modification factor that accounts for the weight of the concrete |

ξ | size effect factor from Bažant and Kim [35] |

ρ | longitudinal reinforcement steel ratio |

σ_{t} | SFRC tensile stress |

(σ_{t})_{avg} | SFRC average tensile stress |

τ_{max} | maximum bond strength of fiber-matrix interface |

ϕ | strength reduction factor for ACI 318-14 [5] |

ψ | size effect factor from Imam et al. [33] |

ω | reinforcement factor including fiber effect |

θ | shear crack angle |

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**Figure 2.**Dramix 3D steel fibers [55].

**Figure 4.**(

**a**) Sketch of the setup for tensile strength test. (

**b**) Failure of specimen in tensile strength test.

**Figure 6.**(

**a**) Sketch of the setup of the experiment. (

**b**) Cross-section of the beam for shear experiments (all units in mm) [58]. (

**c**) Picture of the setup.

**Figure 10.**Normalized shear stress for ultimate shear capacity vs. fiber volume fraction and vs. fiber factor, predictions and measurements.

**Table 1.**Expressions for predicting the ultimate shear capacity of steel fiber-reinforced concrete (SFRC) beams without stirrups.

Authors | Ref | Expression | Equation |
---|---|---|---|

Lee et al. | [18] | ${V}_{sf}=0.41F{\tau}_{\mathrm{max}}{b}_{w}(d-c)\mathrm{cot}\theta $ | (3) |

$\mathrm{with}\text{}{\tau}_{\mathrm{max}}=0.85\sqrt{{f}_{c}}$ | (4) | ||

${V}_{c}=\mathrm{min}(\frac{0.18\lambda \sqrt{{f}_{c}}}{0.31+0.686{w}_{s}}{b}_{w}(d-c),0.52\sqrt{{f}_{c}}{b}_{w}c)$ | (5) | ||

${V}_{u}={V}_{sf}+{V}_{c}$ | (6) | ||

Imam et al. | [33] | ${V}_{u}=0.6\psi \sqrt[3]{\omega}\left[{f}_{c}^{0.44}+275\sqrt{\frac{\omega}{{(a/d)}^{5}}}\right]{b}_{w}d$ $\begin{array}{l}\mathrm{with}\text{}\omega =\rho (1+4F)\\ \psi =\frac{1+\sqrt{5.08/{d}_{a}}}{\sqrt{1+d/(25{d}_{a})}}\end{array}$ | (7) |

Arslan | [48] | ${V}_{u}=\left(0.2{f}_{c}{}^{2/3}\left(\frac{c}{d}\right)+\sqrt{\rho (1+4F){f}_{c}}\right){\left(\frac{3.0}{a/d}\right)}^{1/3}{b}_{w}d$ | (8) |

Dinh et al. | [37] | ${V}_{c}=0.11{f}_{c}{\beta}_{1}c{b}_{w}=0.13{A}_{s}{f}_{y}$ $\mathrm{with}\text{}{\beta}_{1}\text{}\mathrm{from}\text{}\mathrm{Whitney}\u2019\mathrm{s}\text{}\mathrm{stress}\text{}\mathrm{block}$ | (9) |

${V}_{sf}={({\sigma}_{t})}_{avg}{b}_{w}(d-c)\mathrm{cot}(\theta )$ ${({\sigma}_{t})}_{avg}\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{average}\text{}\mathrm{tensile}\text{}\mathrm{stress}\text{}\mathrm{of}\text{}\mathrm{SFRC}$ | (10) | ||

${V}_{u}={V}_{sf}+{V}_{c}$ | (11) | ||

Yakoub | [34] | ${V}_{u}=\left[0.83\xi \sqrt[3]{\rho}\left(\sqrt{{f}_{c}}+249.28\sqrt{\frac{\rho}{{(a/d)}^{5}}}\right)+0.162F\sqrt{{f}_{c}}\right]{b}_{w}d$ $\mathrm{with}\text{}\xi =1/\sqrt{1+d/(25{d}_{a})}$ | (12) |

${V}_{u}=\beta \sqrt{{f}_{c}}(1+0.70F){b}_{w}d$ $\mathrm{with}\text{}\beta =\frac{0.4}{1+1500{\epsilon}_{x}}\frac{1300}{1000+{s}_{xe}}$ $\begin{array}{l}{\epsilon}_{x}=\frac{M/{d}_{v}+V}{2{E}_{s}{A}_{s}}\\ {d}_{v}=\mathrm{max}(0.9d,0.72h)\\ {s}_{xe}=\frac{35{s}_{x}}{16+{d}_{a}}\ge 0.85{s}_{x}\end{array}$ | (13) | ||

Mansur et al. | [39] | ${V}_{sf}={\sigma}_{t}{b}_{w}d$ $\mathrm{with}\text{}{\sigma}_{t}=0.68\sqrt{{f}_{c}}$ | (14) |

${V}_{c}=\left(0.16\sqrt{{f}_{c}}+17.2\frac{\rho Vd}{M}\right){b}_{w}d\overline{)>}(0.29\sqrt{{f}_{c}}){b}_{w}d$ | (15) | ||

${V}_{u}={V}_{sf}+{V}_{c}$ | (16) | ||

Narayanan and Darwish | [30] | ${V}_{u}=\left[e\left(0.24{f}_{sp}+80\rho \frac{d}{a}\right)+{v}_{b}\right]{b}_{w}d$ $\begin{array}{l}e=1\text{}\mathrm{when}\text{}\frac{a}{d}\text{}2.8\\ e=2.8\frac{d}{a}\text{}\mathrm{when}\text{}\frac{a}{d}\le 2.8\end{array}$ | (17) |

$\mathrm{with}\text{}{f}_{sp}=\frac{{f}_{cuf}}{20-\sqrt{F}}+0.7+1.0\sqrt{F}$ | (18) | ||

Kwak et al. | [13] | ${V}_{u}=\left[3.7e{f}_{sp}^{2/3}{\left(\rho \frac{d}{a}\right)}^{1/3}+0.8{v}_{b}\right]{b}_{w}d$ $\begin{array}{l}e=1\text{}\mathrm{when}\text{}\frac{a}{d}3.4\\ e=3.4\frac{d}{a}\text{}\mathrm{when}\text{}\frac{a}{d}\le 3.4\end{array}$ with f _{sp} from Equation (18) | (19) |

Shin et al. | [41] | ${V}_{u}=\left(0.22{f}_{sp}+217\rho \frac{d}{a}+0.834{v}_{b}\right){b}_{w}d$ with f _{sp} from Equation (18) | (20) |

Ashour et al. | [42] | ${V}_{u}=\left[\left(2.11\sqrt[3]{{f}_{c}}+7F\right){\left(\rho \frac{d}{a}\right)}^{0.333}\right]{b}_{w}d\text{}\mathrm{for}\text{}\frac{a}{d}\ge 2.5$ | (21) |

${V}_{u}=\left[\left(0.7\sqrt{{f}_{c}}+7F\right)\frac{d}{a}+17.2\rho \frac{d}{a}\right]{b}_{w}d$ | (22) | ||

Khuntia et al. | [45] | ${V}_{u}=\left[\left(0.167\alpha +0.25F\right)\sqrt{{f}_{c}}\right]{b}_{w}d$ $\mathrm{with}\text{}\alpha =2.5\frac{d}{a}3\text{}\mathrm{for}\text{}a/d2.5$ $\alpha =1\text{}\mathrm{for}\text{}a/d2.5$ | (23) |

Kara | [47] | ${V}_{u}=\left[{\left(\frac{\rho d}{{c}_{0}{c}_{1}(a/d)}\right)}^{3}+\frac{F{d}^{1/4}}{{c}_{2}}+\sqrt{\frac{{c}_{3}{f}_{c}}{d}}\right]{b}_{w}d$ $\mathrm{with}\text{}{c}_{0}=3.324;{c}_{1}=0.909;{c}_{2}=2.289;{c}_{3}=9.436$ | (24) |

Authors | Ref | Expression | Equation |
---|---|---|---|

Arslan et al. | [49] | ${V}_{cr}=0.6\left(0.2{f}_{c}{}^{2/3}\left(\frac{c}{d}\right)+\sqrt{\rho (1+4F){f}_{c}}\right){\left(\frac{3.0}{a/d}\right)}^{1/3}{b}_{w}d$ | (25) |

Narayanan and Darwish | [30] | ${V}_{cr}=\left(0.24{f}_{sp}+20\rho \frac{d}{a}+0.5F\right){b}_{w}d$ | (26) |

Kwak et al. | [13] | ${V}_{cr}=\left(3{f}_{sp}^{2/3}\sqrt[3]{\rho \frac{d}{a}}\right){b}_{w}d$ | (27) |

Fiber Content (%) | Cement (kg/m^{3}) | Fine Aggregates (kg/m^{3}) | Coarse Aggregates (kg/m^{3}) | Water (kg/m^{3}) | Steel Fibers (kg/m^{3}) | w/cm | Fiber Factor |
---|---|---|---|---|---|---|---|

0.0 | 575 | 875 | 585 | 253 | - | 0.40 | 0.00 |

0.3 | 557 | 848 | 567 | 273 | 23.6 | 0.45 | 0.24 |

0.6 | 555 | 845 | 565 | 272 | 47.1 | 0.45 | 0.48 |

0.9 | 538 | 820 | 548 | 291 | 68.7 | 0.50 | 0.72 |

1.2 | 508 | 792 | 518 | 319 | 94.4 | 0.55 | 0.96 |

**Table 4.**Steel fiber properties [55].

Property | Value |
---|---|

Length | 60 mm |

Diameter | 0.75 mm |

Tensile strength | 1225 MPa |

Modulus of Elasticity | 210,000 MPa |

Shape | hooked-end |

Property | Value |
---|---|

Nominal diameter | 16 mm |

Yield Strength | 452 MPa |

Ultimate Strength | 601 MPa |

Modulus of Elasticity | 176,667 MPa |

Fiber Content (%) | Compressive Strength (MPa) | Flexural Stress at First Peak (MPa) | Deflection at First Peak (mm) | Peak Flexural Stress (MPa) | Peak Deflection (mm) | Tension Stiffening Capacity |
---|---|---|---|---|---|---|

0.0 | 20.6 | - | - | 2.88 | 0.600 | - |

0.3 | 33.0 | 1.77 * | 1.260 * | 2.82 | 1.820 | 1.25 * |

0.6 | 27.8 | 2.86 | 0.637 | 5.39 | 3.676 | 1.88 |

0.9 | 29.1 | 3.38 | 0.857 | 6.00 | 2.103 | 1.78 |

1.2 | 30.3 | 5.35 | 1.024 | 6.16 | 1.942 | - |

Fiber Content (%) | Maximum V_{u} [Equation] (kN) | Associated Load (kN) | Minimum M_{n} [Equation] (kN-m) | Associated Load (kN) |
---|---|---|---|---|

0.0 | 21.3 [(12)] | 42.6 | 10.9 [(28) and (29)] | 76.4 |

0.3 | 24.5 [(8)] | 49.0 | 10.9 [(28)] | 76.4 |

0.6 | 27.8 [(8)] | 55.6 | 10.9 [(28)] | 76.4 |

0.9 | 30.6 [(8)] | 61.2 | 10.9 [(28)] | 76.4 |

1.2 | 33.1 [(8)] | 66.2 | 10.9 [(28)] | 76.4 |

Specimen ID | Fiber Content (%) | P_{cr} (kN) | P_{u} (kN) | V_{u} (kN) | $\frac{{\mathit{V}}_{\mathit{u}}}{\left({\mathit{b}}_{\mathit{w}}\mathit{d}\sqrt{{\mathit{f}}_{\mathit{c}}}\right)}$ | δ_{u} (mm) | Failure Mode |
---|---|---|---|---|---|---|---|

VF0.0.1 | 0.0 | 45.74 | 70.1 | 35.20 | 0.772 | 4.738 | Shear |

VF0.0.2 | 0.0 | 46.77 | 57.0 | 28.65 | 0.628 | 4.235 | Shear |

VF0.3.1 | 0.3 | 47.78 | 61.7 | 31.00 | 0.537 | 3.030 | Shear |

VF0.3.2 | 0.3 | 46.99 | 66.8 | 33.55 | 0.581 | 1.603 * | Shear |

VF0.6.1 | 0.6 | 54.62 | 68.1 | 34.20 | 0.646 | 2.606 ^{†} | Shear |

VF0.6.2 | 0.6 | 48.20 | 57.7 | 29.00 | 0.547 | 2.372 | Shear |

VF0.9.1 | 0.9 | 48.48 | 62.5 | 31.40 | 0.579 | 4.000 | Shear |

VF0.9.2 | 0.9 | 41.64 | 55.8 | 28.05 | 0.517 | 3.445 | Shear |

VF1.2.1 | 1.2 | 56.50 | 68.1 | 34.20 | 0.619 | 1.919 ^{‡} | Shear |

VF1.2.2 | 1.2 | 57.88 | 75.2 | 37.75 | 0.683 | 4.000 | Shear + Flexure |

^{†}Deflection at failure in the shear span.

^{‡}Deflection at failure in the shear span.

**Table 9.**Comparison between experimental results and prediction of ultimate shear capacities of SFRC beams.

Authors | Equation | Average Tested/Predicted | Standard Deviation | Coefficient of Variation |
---|---|---|---|---|

Lee et al. | (6) | 1.864 | 0.499 | 0.268 |

Imam et al. | (7) | 1.839 | 0.577 | 0.314 |

Arslan | (8) | 1.244 | 0.373 | 0.230 |

Dinh et al. | (11) | 1.701 | 0.932 | 0.548 |

Yakoub | (12) | 1.289 | 0.214 | 0.166 |

(13) | 1.764 | 0.343 | 0.195 | |

Mansur et al. | (16) | 1.978 | 0.795 | 0.402 |

Narayanan and Darwish | (17) | 1.301 | 0.432 | 0.332 |

Kwak et al. | (19) | 1.209 | 0.421 | 0.348 |

Shin et al. | (20) | 0.744 | 0.113 | 0.152 |

Ashour et al. | (21) | 1.476 | 0.493 | 0.334 |

(22) | 1.351 | 0.603 | 0.446 | |

Khuntia et al. | (23) | 2.394 | 1.081 | 0.452 |

Kara | (24) | 1.432 | 0.420 | 0.294 |

**Table 10.**Comparison between experimental results and prediction of inclined cracking capacities of SFRC beams.

Authors | Equation | Average Tested/Predicted | Standard Deviation | Coefficient of Variation |
---|---|---|---|---|

Arslan | (25) | 1.579 | 0.417 | 0.264 |

Narayanan and Darwish | (26) | 2.096 | 0.598 | 0.255 |

Kwak et al. | (27) | 1.661 | 0.255 | 0.154 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Torres, J.A.; Lantsoght, E.O.L. Influence of Fiber Content on Shear Capacity of Steel Fiber-Reinforced Concrete Beams. *Fibers* **2019**, *7*, 102.
https://doi.org/10.3390/fib7120102

**AMA Style**

Torres JA, Lantsoght EOL. Influence of Fiber Content on Shear Capacity of Steel Fiber-Reinforced Concrete Beams. *Fibers*. 2019; 7(12):102.
https://doi.org/10.3390/fib7120102

**Chicago/Turabian Style**

Torres, Juan Andres, and Eva O.L. Lantsoght. 2019. "Influence of Fiber Content on Shear Capacity of Steel Fiber-Reinforced Concrete Beams" *Fibers* 7, no. 12: 102.
https://doi.org/10.3390/fib7120102