Flexural and direct tensile strength ratio for concrete unusual cross-sections

Fernandes Neto, José Anchiêta Damasceno; Haach, Vladimir Guilherme

doi:10.1590/S1983-41952023000100004

José Anchiêta Damasceno Fernandes Neto

Universidade de São Paulo – USP, Escola de Engenharia de São Carlos, Departamento de Engenharia de Estruturas, São Carlos, SP, Brasil

https://orcid.org/0000-0002-1516-3844

Vladimir Guilherme Haach

Universidade de São Paulo – USP, Escola de Engenharia de São Carlos, Departamento de Engenharia de Estruturas, São Carlos, SP, Brasil

https://orcid.org/0000-0002-9501-4450

Corresponding author: José Anchiêta Damasceno Fernandes Neto.E-mail: anchietafernandes@usp.br

Conflict of interest: Nothing to declare.

Author contributions: JADFN: conceptualization, methodology, validation, data curation, formal analysis, writing – original draft; VGH: conceptualization, methodology, project administration, supervision, writing – review & editing.

Editors: Luís Oliveira Santos, Guilherme Aris Parsekian.

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Figure 1
Plain concrete behavior under tensile loading: (a) stress-strain relationship, (b) moment-curvature relationship, and (c-e) stress-strain distribution diagrams along the uncracked and cracked sections.

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Figure 2
(a) Idealized stress-strain relationship in tension and stress distribution diagrams for (b) perfectly brittle material and (c) perfectly ductile material.

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Figure 3
(a) Stress-strain relationship for concrete in tension and (b) stress-strain distribution diagram on the uncracked ligament.

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Figure 4
Geometry of the cross-sections of precast concrete groups (dimensions in cm).

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Figure 5
Mechanical behavior of concrete: (a) Compression and (b) tension.

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Figure 6
Comparison between theoretical and experimental results.

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Figure 7
Variation in α_fl with ultimate tensile strain for BCS-1 and FLS-3.

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Figure 8
Variation in α_fl with compressive strength for BCS Group.

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Figure 9
Variation in α_fl with compressive strength for FLS Group.

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Figure 10
Variation in α_fl with compressive strength for RFS Group.

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Figure 11
Variation in α_fl with compressive strength for BRS Group.

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Figure 12
Comparison of α_fl obtained by different prediction models.

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Figure 13
α_fl for rectangular, I and T beams: Theoretical results vs. ABNT NBR 6118 [⁹9 Associação Brasileira de Normas Técnicas, Projeto de estruturas de concreto: Procedimento, ABNT NBR 6118, 2014.].

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Figure 7
Variation in α_fl with ultimate tensile strain for BCS-1 and FLS-3.

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Figure 8
Variation in α_fl with compressive strength for BCS Group.

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Figure 9
Variation in α_fl with compressive strength for FLS Group.

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Figure 10
Variation in α_fl with compressive strength for RFS Group.

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Figure 11
Variation in α_fl with compressive strength for BRS Group.

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Figure 12
Comparison of α_fl obtained by different prediction models.

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Figure 13
α_fl for rectangular, I and T beams: Theoretical results vs. ABNT NBR 6118 [⁹9 Associação Brasileira de Normas Técnicas, Projeto de estruturas de concreto: Procedimento, ABNT NBR 6118, 2014.].

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Table 1
Summary of the mechanical properties of concrete.

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Table 2
Variation in α_fl according to aggregate type for f_ck = 40 MPa.

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Table 3
Summary of the expressions for α_fl.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

Property	f_ck ≤ 50 MPa	f_ck > 50 MPa
f_cd	$f_{ck} {/γ}_{c}$	$f_{ck} {/γ}_{c}$
f_ctd	$0 {.21f}_{ck}^{2/3} {/γ}_{c}$	$0 .7 [2 .12ln (1 + 0 {.11f}_{ck})] {/γ}_{c}$
E	${5600α}_{e} f_{ck}^{1/2}$	${21500α}_{e} {(0 {.1f}_{ck} + 1 .25)}^{1/3}$
Ɛ_c2	2.0‰	$2 .0‰ + 0 .085‰ {(f_{ck} - 50)}^{0 .53}$
Ɛ_cu	3.5‰	$2 .6‰ + 35‰ {[({90 - f}_{ck}) /100]}^{4}$
n	2.0	$1 .4 + 23 .4 {[({90 - f}_{ck}) /100]}^{4}$
Ɛ_pt	${βf}_{ctd} /E$	${βf}_{ctd} /E$
Ɛ_ut	$10 ε_{pt}$	$10 ε_{pt}$

Group	Section	Aggregate type				Mean	CV (%)
Group	Section	Basalt or Diabase	Granite or Gneiss	Limestone	Sandstone	Mean	CV (%)
BCS	BCS-1	1.46	1.51	1.54	1.61	1.53	4.10
	BCS-2	1.68	1.75	1.79	1.87	1.77	4.48
	BCS-3	1.40	1.45	1.46	1.52	1.46	3.38
	BCS-4	1.43	1.46	1.50	1.54	1.48	3.23
	BCS-5	1.19	1.22	1.23	1.26	1.23	2.36
	BCS-6	1.40	1.44	1.47	1.52	1.46	3.47
	BCS-7	1.38	1.41	1.42	1.46	1.42	2.33
	BCS-8	1.37	1.41	1.43	1.49	1.43	3.51
FLS	FLS-1	1.26	1.29	1.31	1.33	1.30	2.30
	FLS-2	1.46	1.51	1.54	1.61	1.53	4.10
	FLS-3	1.40	1.45	1.48	1.55	1.47	4.27
	FLS-4	1.38	1.42	1.45	1.51	1.44	3.80
	FLS-5	1.32	1.35	1.37	1.42	1.37	3.08
	FLS-6	1.42	1.46	1.47	1.51	1.47	2.52
	FLS-7	1.44	1.48	1.49	1.53	1.49	2.49
	FLS-8	1.08	1.09	1.10	1.12	1.10	1.56
RFS	RFS-1	1.22	1.24	1.26	1.29	1.25	2.38
	RFS-2	1.48	1.54	1.57	1.66	1.56	4.80
	RFS-3	1.51	1.57	1.60	1.68	1.59	4.45
	RFS-4	1.31	1.35	1.38	1.44	1.37	4.00
	RFS-5	1.27	1.31	1.33	1.38	1.32	3.46
	RFS-6	1.53	1.59	1.62	1.70	1.61	4.39
	RFS-7	1.23	1.26	1.27	1.32	1.27	2.95
	RFS-8	1.37	1.42	1.44	1.51	1.44	4.04
BRS	BRS-1	1.12	1.14	1.16	1.19	1.15	2.59
	BRS-2	1.29	1.32	1.34	1.39	1.34	3.15
	BRS-3	1.17	1.21	1.22	1.26	1.22	3.04
	BRS-4	1.13	1.16	1.18	1.22	1.17	3.22
	BRS-5	1.11	1.13	1.14	1.18	1.14	2.58
	BRS-6	1.14	1.17	1.19	1.24	1.19	3.55
	BRS-7	1.18	1.21	1.23	1.27	1.22	3.09
	BRS-8	1.40	1.44	1.46	1.51	1.45	3.15

Model	Ratio α_fl
Model Code [¹⁴14 International Federation for Structural Concrete, fib Model Code 2010, MC2010, 2010.]	$α_{fl} = \frac{1+0 {.06h}^{0 .7}}{0 {.06h}^{0 .7}}$
Müller and Hilsdorf [³¹31 H. S. Müller and H. K. Hilsdorf, “Constitutive relations of structural concrete,” CEB Bullet. Inf., no. 217, pp. 17-65, Apr, 1993.]	$α_{fl} = \frac{[{1 + α}_{mh} {(0 .01h)}^{0 .7}]}{(α_{mh} {(0 .01h)}^{0 .7})} {; with α}_{mh} = 0 .8 + \frac{5}{{(0 {.01l}_{ch})}^{1 .5}} {and l}_{ch} = \frac{G_{f} E}{f_{ct}^{2}}$
ABNT NBR 6118 [⁹9 Associação Brasileira de Normas Técnicas, Projeto de estruturas de concreto: Procedimento, ABNT NBR 6118, 2014.]	$α_{fl} = 1 .5, 1 .3 or 1 .2 for rectangular, I and T beams, respectively .$
Buchaim [²³23 R. Buchaim, “A influência da não-linearidade física do concreto armado na rigidez à flexão e na capacidade de rotação plástica,” Ph.D dissertation, Dept. Eng. Estrut. Fund., Univ. São Paulo, São Paulo, 2001.]	$α_{fl} = 1 + 2η [\frac{3 - 6η + 3 (1 - B) η^{2} {+ 2Bη}^{3}}{3 - 6η + (B + 3) η^{2}}]; with η = \frac{a_{d}}{h} and B = \frac{f_{ct}^{2} L}{{2G}_{f} E}$
Rokugo et al. [³²32 K. Rokugo, Y. H. Katoh, and W. Koyanagi, "Fracture mechanics approach to evaluation of flexural strength of concrete," ACI Mater. J., vol. 92, no. 5, pp. 561–566, Jan 1993.]	$α_{fl} = \frac{[1 + 0 .85 + 4 .5 ({h/l}_{ch})]}{[0 .85 + 4 .5 ({h/l}_{ch})]} {; with l}_{ch} = \frac{G_{f} E}{f_{ct}^{2}}$

[1] Corresponding author: José Anchiêta Damasceno Fernandes Neto.E-mail: anchietafernandes@usp.br

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Flexural and direct tensile strength ratio for concrete unusual cross-sections

Relação entre as resistências à tração na flexão e direta para seções transversais não usuais de concreto

ABSTRACT