Evaluation of the replacement of minimum shear reinforcement by steel fibers in reinforced concrete beams

Bono, Giuliana Furtado Franca; Bono, Gustavo; Santos Júnior, Marinaldo dos

doi:10.1590/S1983-41952024000400005

Giuliana Furtado Franca Bono

conceptualization

data curation

formal analysis

writing

supervision

Universidade Federal de Pernambuco - UFPE, Programa de Pós-graduação em Engenharia Civil e Ambiental, Caruaru, PE, Brasil

https://orcid.org/0000-0002-3668-135X

Gustavo Bono

conceptualization

data curation

formal analysis

writing

supervision

Universidade Federal de Pernambuco - UFPE, Programa de Pós-graduação em Engenharia Civil e Ambiental, Caruaru, PE, Brasil

https://orcid.org/0000-0003-4666-8703

Marinaldo dos Santos Júnior

conceptualization

data curation

dates acquisition

writing

Instituto Federal da Paraíba - IFPB, Princesa Isabel, PB, Brasil

https://orcid.org/0000-0003-3515-3667

Corresponding author: Giuliana Furtado Franca Bono. E-mail: giuliana.franca@ufpe.br

Conflict of interest: Nothing to declare.

Editors: Leandro Trautwein, Mario Pimentel, Mauro de Vasconcellos Real.

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Figure 1
Distribution of parameters in database [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]: (a) beam width and (b) beam height.

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Figure 5
Distribution of parameters in database [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]: (a) fiber volume and (b) aspect factor.

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Figure 2
Distribution of parameters in database [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]: (a) effective height and (b) span length.

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Figure 3
Distribution of parameters in database [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]: (a) shear span to depth ratio and (b) longitudinal reinforcement ratio.

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Figure 4
Distribution of parameters in database [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]: (a) maximum aggregate size and (b) compressive strength of concrete.

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Figure 6
Relationship between experimental and theoretical values of the ultimate shear force: (a) Singh and Jain [⁵³53 B. Singh and K. Jain, “An appraisal of steel fibers as minimum shear reinforcement in concrete beams,” ACI Struct. J., vol. 111, no. 5, pp. 1191-1202, Sep. 2014, http://dx.doi.org/10.14359/51686969.
http://dx.doi.org/10.14359/51686969... ], (b) Sahoo and Sharma [⁵⁴54 D. R. Sahoo and A. Sharma, “Effect of steel fiber content on behavior of concrete beams with and without stirrups,” ACI Struct. J., vol. 111, no. 5, pp. 1157-1166, Sep. 2014, http://dx.doi.org/10.14359/51686821.
http://dx.doi.org/10.14359/51686821... ], (c) Dinh et al. [⁹⁴94 H. H. Dinh, G. J. Parra-Montesinos, and J. K. Wight, “Shear strength model for steel fiber reinforced concrete beams without stirrup reinforcement,” J. Struct. Eng., vol. 137, no. 10, pp. 1039-1051, Dec. 2011., http://dx.doi.org/10.1061/(ASCE)ST.1943-541X.0000362.
http://dx.doi.org/10.1061/(ASCE)ST.1943-... ], (d) Kwak et al. [²2 Y. K. Kwak, M. O. Eberhard, W. S. Kim, and J. Kim, “Shear strength of steel fiber-reinforced concrete beams without stirrups,” ACI Struct. J., vol. 49, no. 4, pp. 530-538, Jul. 2002, http://dx.doi.org/10.14359/12122.
http://dx.doi.org/10.14359/12122... ], (e) Sharma [⁴⁴44 A. K. Sharma, “Shear strength of steel fiber reinforced concrete beams,” ACI J. Proc., vol. 83, no. 4, pp. 624-628, Jul. 1986, http://dx.doi.org/10.14359/10559.
http://dx.doi.org/10.14359/10559... ], (f) Sarveghadi et al. [⁹⁵95 M. Sarveghadi, A. H. Gandomi, H. Bolandi, and A. H. Alavi, “Development of prediction models for shear strength of SFRCB using a machine learning approach,” Neural Comput. Appl., vol. 31, pp. 2085-2094, Aug. 2015, http://dx.doi.org/10.1007/s00521-015-1997-6.
http://dx.doi.org/10.1007/s00521-015-199... ], (g) Imam et al. [⁹⁶96 M. Imam, L. Vandewalle, F. Mortelmans, and D. Van Gemert, “Shear domain of fibre-reinforced high-strength concrete beams,” Eng. Struct., vol. 19, no. 9, pp. 738-747, Sep. 1997, http://dx.doi.org/10.1016/S0141-0296(96)00150-2.
http://dx.doi.org/10.1016/S0141-0296(96)... ], (h) Arslan [⁹³93 G. Arslan, “Shear strength of Steel Fiber Reinforced Concrete (SFRC) slender beams,” KSCE J. Civ. Eng., vol. 18, pp. 587-594, Mar. 2014, http://dx.doi.org/10.1007/s12205-014-0320-x.
http://dx.doi.org/10.1007/s12205-014-032... ], (i) DAfStB [¹⁹19 DAfStB, Deutscher Ausschuss für Stahlbeton (DAfStb): Richtlinie Stahlfaserbeton, Berlin, Germany, 2012.], (j) fib Model Code [²⁰20 fib, Model Code 2010: Final Draft, fib Model Code for Concrete Structures 2010. International Federation for Structural Concrete (fib), 2012.], (k) RILEM [²²22 RILEM, “RILEM TC 162-TDF: ‘Test and design methods for steel fibre reinforced concrete’, σ-ε-design method,” Mater. Struct., vol. 36, pp. 560-567, Oct. 2003.], (l) Linear regression [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.] and (m) Weighted linear regression [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.].

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Figure 7
Details of the cross section of the beam.

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Figure 8
Abacus B2.3 [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.] for steel fiber volume in beam.

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Figure 9
Influence of

f_{c}

,

ρ

,

V_{f}

in

V_{d}

and the amount of steel fibers required to replace the minimum shear reinforcement in SFRC beams: (a)

ρ = 1.0 %

, (b)

ρ = 1.50 %

.

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Figure 7
Details of the cross section of the beam.

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Figure 8
Abacus B2.3 [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.] for steel fiber volume in beam.

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Figure 9
Influence of

f_{c}

,

ρ

,

V_{f}

in

V_{d}

and the amount of steel fibers required to replace the minimum shear reinforcement in SFRC beams: (a)

ρ = 1.0 %

, (b)

ρ = 1.50 %

.

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Table 1
Ranges of parameters in database [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.].

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Table 2
Models for predicting shear strength in SFRC beams.

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Table 3
Characteristics of the groups according to the type of steel fiber used and the

a / d

[⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.].

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Table 4
Statistical analysis of shear strength prediction models in SFRC beams.

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(49)

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(50)

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(51)

Figure 6 Relationship between experimental and theoretical values of the ultimate shear force: (a) Singh and Jain [⁵³53 B. Singh and K. Jain, “An appraisal of steel fibers as minimum shear reinforcement in concrete beams,” ACI Struct. J., vol. 111, no. 5, pp. 1191-1202, Sep. 2014, http://dx.doi.org/10.14359/51686969.
http://dx.doi.org/10.14359/51686969... ], (b) Sahoo and Sharma [⁵⁴54 D. R. Sahoo and A. Sharma, “Effect of steel fiber content on behavior of concrete beams with and without stirrups,” ACI Struct. J., vol. 111, no. 5, pp. 1157-1166, Sep. 2014, http://dx.doi.org/10.14359/51686821.
http://dx.doi.org/10.14359/51686821... ], (c) Dinh et al. [⁹⁴94 H. H. Dinh, G. J. Parra-Montesinos, and J. K. Wight, “Shear strength model for steel fiber reinforced concrete beams without stirrup reinforcement,” J. Struct. Eng., vol. 137, no. 10, pp. 1039-1051, Dec. 2011., http://dx.doi.org/10.1061/(ASCE)ST.1943-541X.0000362.
http://dx.doi.org/10.1061/(ASCE)ST.1943-... ], (d) Kwak et al. [²2 Y. K. Kwak, M. O. Eberhard, W. S. Kim, and J. Kim, “Shear strength of steel fiber-reinforced concrete beams without stirrups,” ACI Struct. J., vol. 49, no. 4, pp. 530-538, Jul. 2002, http://dx.doi.org/10.14359/12122.
http://dx.doi.org/10.14359/12122... ], (e) Sharma [⁴⁴44 A. K. Sharma, “Shear strength of steel fiber reinforced concrete beams,” ACI J. Proc., vol. 83, no. 4, pp. 624-628, Jul. 1986, http://dx.doi.org/10.14359/10559.
http://dx.doi.org/10.14359/10559... ], (f) Sarveghadi et al. [⁹⁵95 M. Sarveghadi, A. H. Gandomi, H. Bolandi, and A. H. Alavi, “Development of prediction models for shear strength of SFRCB using a machine learning approach,” Neural Comput. Appl., vol. 31, pp. 2085-2094, Aug. 2015, http://dx.doi.org/10.1007/s00521-015-1997-6.
http://dx.doi.org/10.1007/s00521-015-199... ], (g) Imam et al. [⁹⁶96 M. Imam, L. Vandewalle, F. Mortelmans, and D. Van Gemert, “Shear domain of fibre-reinforced high-strength concrete beams,” Eng. Struct., vol. 19, no. 9, pp. 738-747, Sep. 1997, http://dx.doi.org/10.1016/S0141-0296(96)00150-2.
http://dx.doi.org/10.1016/S0141-0296(96)... ], (h) Arslan [⁹³93 G. Arslan, “Shear strength of Steel Fiber Reinforced Concrete (SFRC) slender beams,” KSCE J. Civ. Eng., vol. 18, pp. 587-594, Mar. 2014, http://dx.doi.org/10.1007/s12205-014-0320-x.
http://dx.doi.org/10.1007/s12205-014-032... ], (i) DAfStB [¹⁹19 DAfStB, Deutscher Ausschuss für Stahlbeton (DAfStb): Richtlinie Stahlfaserbeton, Berlin, Germany, 2012.], (j) fib Model Code [²⁰20 fib, Model Code 2010: Final Draft, fib Model Code for Concrete Structures 2010. International Federation for Structural Concrete (fib), 2012.], (k) RILEM [²²22 RILEM, “RILEM TC 162-TDF: ‘Test and design methods for steel fibre reinforced concrete’, σ-ε-design method,” Mater. Struct., vol. 36, pp. 560-567, Oct. 2003.], (l) Linear regression [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.] and (m) Weighted linear regression [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.].

Parameter	Minimum	Maximum
$b_{w}$ (mm)	100	310
$h$ (mm)	152	1000
$d$ (mm)	127	923
$L$ (mm)	762	5538
$a / d$	0.58	5
$ρ$ (%)	0.99	4.10
$d_{a}$ (mm)	2	22
$f_{c}$ (MPa)	9.77	49.9
$V_{f}$ (%)	0.20	3.0
$l_{f}$ (mm)	7.05	60
$d_{f}$ (mm)	0.15	1.11
$l_{f} / d_{f}$	25	101.6

Ref.	Expression
Model 1 - Singh and Jain [⁵³53 B. Singh and K. Jain, “An appraisal of steel fibers as minimum shear reinforcement in concrete beams,” ACI Struct. J., vol. 111, no. 5, pp. 1191-1202, Sep. 2014, http://dx.doi.org/10.14359/51686969. http://dx.doi.org/10.14359/51686969... ]	$V_{u} = V_{C C} + V_{F R C}$	(1)
	$V_{C C} = 0.13 A_{s} f_{y}$	(2)
	$V_{F R C} = 0.5 τ D_{f} l_{f} / d_{f} b_{w} (d - c) \cot α$	(3)
	$τ = \{\begin{matrix} 0.85 \sqrt{f_{c}} \to F o r h o o k e d f i b e r s \\ 0.75 \sqrt{f_{c}} \to F o r c r i m p e d f i b e r s \end{matrix}$	(4)
Model 2 - Sahoo and Sharma [⁵⁴54 D. R. Sahoo and A. Sharma, “Effect of steel fiber content on behavior of concrete beams with and without stirrups,” ACI Struct. J., vol. 111, no. 5, pp. 1157-1166, Sep. 2014, http://dx.doi.org/10.14359/51686821. http://dx.doi.org/10.14359/51686821... ]	$V_{u} = (0.251 + 0.173 V_{f} + 0.069 {V_{f}}^{2}) \sqrt{f_{c}} b_{w} d$ (d < 300 mm)	(5)
	$V_{u} = (0.202 + 0.377 V_{f} - 0.113 {V_{f}}^{2}) \sqrt{f_{c}} b_{w} d$ ( d ≥ 300 mm)	(6)
Model 3 - Dinh et al. [⁵²52 H. H. Dinh, G. J. Parra-Montesinos, and J. K. Wight, “Shear behavior of steel fiber-reinforced concrete beams without stirrup reinforcement,” ACI Struct. J., vol. 107, no. 5, pp. 597-606, Sep. 2010.]	$V_{u} = V_{C C} + V_{F R C}$	(7)
	$V_{C C} = 0.13 A_{s} f_{y}$	(8)
	$V_{F R C} = {(σ_{t})}_{a v g} b_{w} (d - c) \cot α$	(9)
	$c = \frac{A_{s} f_{y}}{k_{1} k_{3} f_{c} b_{w}}$	(10)
	$k_{1} k_{3} = 0.85 β_{1}$	(11)
	$β_{1} = \{\begin{matrix} 0.85 \to f_{c} \leq 27.6 M P a \\ 0.65 \to f_{c} \geq 55.1 M P a \\ I n t e r p o l a t i o n \to {27.6 M P a < f}_{c} < 55.1 M P a \end{matrix}$	(12)
Model 4 - Kwak et al. [²2 Y. K. Kwak, M. O. Eberhard, W. S. Kim, and J. Kim, “Shear strength of steel fiber-reinforced concrete beams without stirrups,” ACI Struct. J., vol. 49, no. 4, pp. 530-538, Jul. 2002, http://dx.doi.org/10.14359/12122. http://dx.doi.org/10.14359/12122... ]	$V_{u} = [3.7 e f_{s p f c}^{2 / 3} {(ρ \frac{d}{a})}^{1 / 3} + 0.8 υ_{b}] b_{w} d$	(13)
	$f_{s p f c} = \frac{f_{c u f}}{(20 - \sqrt{F})} + 0.7 + \sqrt{F}$	(14)
	$e = \{\begin{matrix} 1 \to \frac{a}{d} > 3.4 \\ 3.4 \frac{d}{a} \to \frac{a}{d} \leq 3.4 \end{matrix}$	(15)
	$υ_{b} = 0.41 τ F$	(16)
	$F = (l_{f} / d_{f}) V_{f} β$	(17)
Model 5 - Sharma [⁴⁴44 A. K. Sharma, “Shear strength of steel fiber reinforced concrete beams,” ACI J. Proc., vol. 83, no. 4, pp. 624-628, Jul. 1986, http://dx.doi.org/10.14359/10559. http://dx.doi.org/10.14359/10559... ]	$V_{u} = [\frac{2}{3} f_{c t} {(\frac{d}{a})}^{0.25}] b_{w} d$	(18)
Model 6 - Sarveghadi et al. [⁹⁵95 M. Sarveghadi, A. H. Gandomi, H. Bolandi, and A. H. Alavi, “Development of prediction models for shear strength of SFRCB using a machine learning approach,” Neural Comput. Appl., vol. 31, pp. 2085-2094, Aug. 2015, http://dx.doi.org/10.1007/s00521-015-1997-6. http://dx.doi.org/10.1007/s00521-015-199... ]	$V_{u} = [ρ + \frac{ρ}{υ_{b}} + \frac{1}{\frac{a}{d}} (\frac{ρ f_{t} (ρ + 2) (f_{t} \frac{a}{d} - \frac{3}{υ_{b}})}{\frac{a}{d}} + f_{t}) + υ_{b}] b_{w} d$	(19)
	$f_{t} = 0.79 \sqrt{f_{c}}$	(20)
Model 7 - Imam et al. [⁹⁶96 M. Imam, L. Vandewalle, F. Mortelmans, and D. Van Gemert, “Shear domain of fibre-reinforced high-strength concrete beams,” Eng. Struct., vol. 19, no. 9, pp. 738-747, Sep. 1997, http://dx.doi.org/10.1016/S0141-0296(96)00150-2. http://dx.doi.org/10.1016/S0141-0296(96)... ]	$V_{u} = [0.6 ψ \sqrt[3]{ω} ({(f_{c})}^{0.44} + 275 \sqrt{ω / {(\frac{a}{d})}^{5}})] b_{w} d$	(21)
	$ψ = (1 + \sqrt{5.08 / d_{a}}) / \sqrt{1 + (d / 25 d_{a})}$	(22)
	$ω = ρ (1 + 4 F)$	(23)
Model 8 - Arslan [⁹³93 G. Arslan, “Shear strength of Steel Fiber Reinforced Concrete (SFRC) slender beams,” KSCE J. Civ. Eng., vol. 18, pp. 587-594, Mar. 2014, http://dx.doi.org/10.1007/s12205-014-0320-x. http://dx.doi.org/10.1007/s12205-014-032... ]	$V_{u} = [({0.2 (f_{c})}^{2 / 3} \frac{c}{d} + \sqrt{ρ (1 + 4 F) f_{c}}) \sqrt[3]{\frac{3}{\frac{a}{d}}}] b_{w} d$	(24)
	${(\frac{c}{d})}^{2} + (\frac{600 ρ}{f_{c}}) (\frac{c}{d}) - \frac{600 ρ}{f_{c}} = 0$	(25)
Model 9 - DafStB [¹⁹19 DAfStB, Deutscher Ausschuss für Stahlbeton (DAfStb): Richtlinie Stahlfaserbeton, Berlin, Germany, 2012.]	$V_{R d, c t}^{f} = V_{R d, c t} + V_{R d, c f}$	(26)
	$V_{R d, c t} = \frac{C_{R d, c}}{γ_{c}} k {(100 ρ f_{c k})}^{1 / 3} b_{w} d$	(27)
	$V_{R d, c f} = \frac{α_{c}^{f} f_{c t R, u}^{f} b_{w} h}{γ_{c t}^{f}}$	(28)
	$f_{c t R, u}^{f} = k_{F}^{f} k_{G}^{f} 0.37 f_{c f I k, L 2}^{f}$	(29)
	$k_{G}^{f} = 1.0 + 0.5 A_{c t}^{f} \leq 1.7$	(30)
	$A_{c t}^{f} = b_{w} m i n . (d; 1.5 m)$	(31)
	$k = 1 + \sqrt{200 m m / d}$	(32)
Model 10 - fib model code [²⁰20 fib, Model Code 2010: Final Draft, fib Model Code for Concrete Structures 2010. International Federation for Structural Concrete (fib), 2012.]	$V_{R d, f} = \frac{C_{R d, c}}{γ_{c}} k {(100 ρ (1 + 7.5 \frac{f_{F t u k}}{f_{c t k}}) f_{c k})}^{1 / 3} b_{w} d$	(33)
	$f_{F t u k} = 0.06 f_{R 1} + 0.3 f_{R 3}$	(34)
	$f_{c t k} = 0.3 {(f_{c k})}^{2 / 3}$ , for $f_{c k} \leq 50 M P a$	(35)
Model 11 - RILEM [²²22 RILEM, “RILEM TC 162-TDF: ‘Test and design methods for steel fibre reinforced concrete’, σ-ε-design method,” Mater. Struct., vol. 36, pp. 560-567, Oct. 2003.]	$V_{R d} = V_{c d} + V_{f d}$	(36)
	$V_{c d} = 0.12 k {(100 ρ f_{c k})}^{1 / 3} b_{w} d$	(37)
	$V_{f d} = 0.7 k_{f} k τ_{f d} b_{w} h$	(38)
	$k_{f} = 1.0 + n (\frac{h_{f}}{b_{w}}) (\frac{h_{f}}{d}) \leq 1.5$	(39)
	$n = \frac{b_{f} - b_{w}}{h_{f}} \leq 3 a n d n \leq \frac{3 b_{w}}{h_{f}}$	(40)
	$τ_{f d} = 0.12 f_{R k, 4}$	(41)
Model 12 - Linear Regression [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]	$V_{u} = (1.001 + 14.551 ρ - 0.208 \frac{a}{d} - 0.233 ρ_{f} - 0.051 V_{f} - 0.093 d_{f} + 0.219 F) \sqrt{f_{c}} b_{w} d$	(42)
Model 13 - Weighted Linear Regression (WLR) [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]	$V_{u} = 474.41 + 2157.81 ρ - 83.49 \frac{a}{d} + 1.61 f_{c} + 2.073 l_{f} - 70.72 V_{f} - 32.46 d_{f} - 1.94 \frac{l_{f}}{d_{f}} - 0.162 f_{f} + 106.07 F$ (For hooked fibers end and $a / d < 3.0$ )	(43)
	$V_{u} = - 245.20 - 59.79 \frac{a}{d} + 3.90 f_{c} + 33.17 V_{f} + 0.218 f_{f}$ (For crimped fibers and $a / d < 3.0$ )	(44)
	$V_{u} = 59.97 - 3769.81 ρ - 6.36 \frac{a}{d} + 8.99 f_{c} + 1.44 l_{f} - 6.89 V_{f} - 162.24 d_{f} - 1.48 \frac{l_{f}}{d_{f}} - 0.024 f_{f} + 147.81 F$ (For other types of fibers and $a / d < 3.0$ )	(45)
	$V_{u} = 907.90 - 6032.88 ρ - 154.46 \frac{a}{d} + 4.26 f_{c} + 2.13 l_{f} - 91.87 V_{f} - 199.69 d_{f} - 1.43 \frac{l_{f}}{d_{f}} - 0.074 f_{f} + 143.22 F$ (For hooked fibers and $a / d \geq 3.0$ )	(46)
	$V_{u} = 694.72 - 7.79 \frac{a}{d} + 1.54 f_{c} + 2.07 l_{f} - 12.97 V_{f} - 1.13 \frac{l_{f}}{d_{f}} - 0.605 f_{f}$ (For crimped fibers and $a / d \geq 3.0$ )	(47)
	$V_{u} = - 2344.49 + 37569.49 ρ - 3.96 \frac{a}{d} - 3.82 f_{c} - 83.88 l_{f} - 92.22 V_{f} + 5043.99 d_{f} + 20.78 \frac{l_{f}}{d_{f}} + 0.081 f_{f} + 432.59 F$ (For other types of fibers and $a / d \geq 3.0$ )	(48)

Groups	$a / d$	Types of Fibers	Number of beams
G1	$< 3.0$	hooked	68 (28.33%)
G2		crimped	10 (4.17%)
G3		other	12 (5.0%)
G4	$\geq 3.0$	hooked	93 (38.75%)
G5		crimped	31 (12.92%)
G6		other	26 (10.83%)

Model	$\frac{V_{u - e x p}}{V_{u - t e o}}$			$S . D .$	$C . V .$	E_mean	$R ²$	$V_{u - t e o}$
Model	Mean	Max	Min	$S . D .$	(%)	(%)	$R ²$	Max (kN)	Min (kN)
Singh and Jain [⁵³53 B. Singh and K. Jain, “An appraisal of steel fibers as minimum shear reinforcement in concrete beams,” ACI Struct. J., vol. 111, no. 5, pp. 1191-1202, Sep. 2014, http://dx.doi.org/10.14359/51686969. http://dx.doi.org/10.14359/51686969... ]	1.26	4.41	0.37	0.59	46.7	27.2	0,659	783.87	16.51
Sahoo and Sharma [⁵⁴54 D. R. Sahoo and A. Sharma, “Effect of steel fiber content on behavior of concrete beams with and without stirrups,” ACI Struct. J., vol. 111, no. 5, pp. 1157-1166, Sep. 2014, http://dx.doi.org/10.14359/51686821. http://dx.doi.org/10.14359/51686821... ]	2.06	8.23	0.95	1.13	54.7	43.8	0,626	364.82	14.60
Dinh et al. [⁹⁴94 H. H. Dinh, G. J. Parra-Montesinos, and J. K. Wight, “Shear strength model for steel fiber reinforced concrete beams without stirrup reinforcement,” J. Struct. Eng., vol. 137, no. 10, pp. 1039-1051, Dec. 2011., http://dx.doi.org/10.1061/(ASCE)ST.1943-541X.0000362. http://dx.doi.org/10.1061/(ASCE)ST.1943-... ]	1.46	5.38	0.65	0.63	43.4	27.7	0,687	543.70	21.30
Kwak et al. [²2 Y. K. Kwak, M. O. Eberhard, W. S. Kim, and J. Kim, “Shear strength of steel fiber-reinforced concrete beams without stirrups,” ACI Struct. J., vol. 49, no. 4, pp. 530-538, Jul. 2002, http://dx.doi.org/10.14359/12122. http://dx.doi.org/10.14359/12122... ]	1.04	1.87	0.33	0.26	24.9	23.0	0,801	840.35	16.62
Sharma [⁴⁴44 A. K. Sharma, “Shear strength of steel fiber reinforced concrete beams,” ACI J. Proc., vol. 83, no. 4, pp. 624-628, Jul. 1986, http://dx.doi.org/10.14359/10559. http://dx.doi.org/10.14359/10559... ]	1.17	3.52	0.54	0.42	36.4	21.7	0,691	718.51	20.81
Sarveghadi et al. [⁹⁵95 M. Sarveghadi, A. H. Gandomi, H. Bolandi, and A. H. Alavi, “Development of prediction models for shear strength of SFRCB using a machine learning approach,” Neural Comput. Appl., vol. 31, pp. 2085-2094, Aug. 2015, http://dx.doi.org/10.1007/s00521-015-1997-6. http://dx.doi.org/10.1007/s00521-015-199... ]	1.05	1.80	0.34	0.28	26.7	23.1	0,837	809.09	16.66
Imam et al. [⁹⁶96 M. Imam, L. Vandewalle, F. Mortelmans, and D. Van Gemert, “Shear domain of fibre-reinforced high-strength concrete beams,” Eng. Struct., vol. 19, no. 9, pp. 738-747, Sep. 1997, http://dx.doi.org/10.1016/S0141-0296(96)00150-2. http://dx.doi.org/10.1016/S0141-0296(96)... ]	1.01	1.91	0.09	0.32	31.3	43.2	0,257	4592.48	18.68
Arslan [⁹³93 G. Arslan, “Shear strength of Steel Fiber Reinforced Concrete (SFRC) slender beams,” KSCE J. Civ. Eng., vol. 18, pp. 587-594, Mar. 2014, http://dx.doi.org/10.1007/s12205-014-0320-x. http://dx.doi.org/10.1007/s12205-014-032... ]	1.14	2.43	0.63	0.29	25.9	17.5	0,805	722.18	18.85
DAfStB [¹⁹19 DAfStB, Deutscher Ausschuss für Stahlbeton (DAfStb): Richtlinie Stahlfaserbeton, Berlin, Germany, 2012.]	1.35	4.42	0.65	0.62	45.9	23.4	0,684	559.96	21.12
Fib [²⁰20 fib, Model Code 2010: Final Draft, fib Model Code for Concrete Structures 2010. International Federation for Structural Concrete (fib), 2012.]	1.37	4.49	0.66	0.63	46.0	25.1	0,686	479.62	22.87
Linear regression [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]	1.05	6.46	0.46	0.45	43.5	22.6	0,801	806.69	6.57
WLR [⁹¹91 M. Santos Jr., “Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado,” Master thesis, Prog. Pós-Grad. Eng. Civil Amb., Univ. Fed. Pernambuco, Caruaru, Brazil, 2021.]	1.01	2.44	-3.63	0.47	46.6	33.4	0,642	375.75	-9.64

[1] Corresponding author: Giuliana Furtado Franca Bono. E-mail: giuliana.franca@ufpe.br

Brasil

Brasil

Evaluation of the replacement of minimum shear reinforcement by steel fibers in reinforced concrete beams

Avaliação da substituição da armadura transversal mínima por fibras de aço em vigas de concreto armado

Abstract