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

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

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

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

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