Computational modeling of plain and steel fiber-reinforced concrete beams without transverse reinforcement

Araújo, Daniel de Lima; Siqueira Filho, Cleiton Rodrigues; Lobo, Fausto Arantes

doi:10.1590/S1983-41952023000300011

Daniel de Lima Araújo

supervision

project administration

funding acquisition

methodology

formal analysis

writing

Universidade Federal de Goiás – UFG, Escola de Engenharia Civil e Ambiental, Goiânia, GO, Brasil

https://orcid.org/0000-0002-6802-4637

Cleiton Rodrigues Siqueira Filho

methodology

formal analysis

investigation

Universidade Federal de Goiás – UFG, Escola de Engenharia Civil e Ambiental, Goiânia, GO, Brasil

https://orcid.org/0009-0006-7938-4925

Fausto Arantes Lobo

methodology

investigation

Universidade Federal de Goiás – UFG, Escola de Engenharia Civil e Ambiental, Goiânia, GO, Brasil

https://orcid.org/0000-0002-5124-2072

Corresponding author: Daniel de Lima Araújo. E-mail: dlaraujo@ufg.br

Conflict of interest: Nothing to declare.

Author contributions: DLA: supervision, project administration, funding acquisition, methodology, formal analysis, writing. CRSF: methodology, formal analysis, investigation. FAL: methodology, investigation.

Editors: Leandro Trautwein, Mauro Real, Mario Pimentel.

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Figure 1
(a) Double-notched push-through test setup (dimensions in mm) and (b) instrumentation.

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Figure 2
Test setup in the pre-cracking stage: (a) frontal view of specimen (dimensions in mm), (b) lateral view of specimen (dimensions in mm), and (c) splitting test.

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Figure 3
Push-off test instrumentation: (a) front and opposite faces and (b) test setup in the push‐off stage.

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Figure 4
Geometry of the double-notched push-through specimen with uncracked planes (dimensions in mm).

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Figure 5
Influence of (a) constant shear retention factor and (b) variable shear retention factor on double-notched push-through test results (F-00-2).

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Figure 6
Relationship of shear stress and crack width in double-notched push-through tests (F-00-1).

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Figure 7
(a) Proposed equations for the variable shear retention factor (β) and (b) shear stress–slip curves obtained using the finite element model and several variable shear retention equations.

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Figure 8
Push-off specimen with pre-cracked shear plane (dimensions in mm).

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Figure 9
(a) Bilinear tension softening diagram for steel fiber-reinforced concrete and (b) load–displacement curve obtained according to ASTM C1609 for F-15-2 mixture.

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Figure 10
Relationship curves for the push-off specimen without steel fibers (F-00-2).

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Figure 11
Relationship curves for the push-off specimen with steel fibers (F-15-2).

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Figure 12
Shear stress versus crack width for push-off specimens with and without steel fibers.

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Figure 13
Geometry of reinforced concrete (RC) beams used for modeling (dimensions in mm).

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Figure 14
Finite element mesh of reinforced concrete beam V-0-0.

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Figure 15
Finite element mesh of reinforced concrete beam 1R.

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Figure 16
Load–displacement curve of the mid-span of the reinforced concrete beam V-0-0 (a/d = 2.17).

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Figure 17
Crack pattern of the reinforced concrete beam V-0-0 (a/d = 2.17).

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Figure 18
Load–displacement curves of the mid-span of the reinforced concrete beam 1R (a/d = 2.66).

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Figure 19
Crack pattern of the reinforced concrete beam 1R (a/d = 2.66).

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Figure 20
Load–displacement curves of the mid-span of the fibrous concrete beam V-1-0 (a/d = 2.17).

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Figure 21
Crack pattern of the fibrous concrete beam V-1-0 (a/d = 2.17).

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Figure 14
Finite element mesh of reinforced concrete beam V-0-0.

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Figure 15
Finite element mesh of reinforced concrete beam 1R.

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Figure 16
Load–displacement curve of the mid-span of the reinforced concrete beam V-0-0 (a/d = 2.17).

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Figure 17
Crack pattern of the reinforced concrete beam V-0-0 (a/d = 2.17).

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Figure 18
Load–displacement curves of the mid-span of the reinforced concrete beam 1R (a/d = 2.66).

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Figure 19
Crack pattern of the reinforced concrete beam 1R (a/d = 2.66).

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Figure 20
Load–displacement curves of the mid-span of the fibrous concrete beam V-1-0 (a/d = 2.17).

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Figure 21
Crack pattern of the fibrous concrete beam V-1-0 (a/d = 2.17).

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Table 1
Mechanical properties of concrete.

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Table 2
Empirical values of variable shear retention factor (β) estimated from double-notched push-through test results (F-00-1).

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Table 3
Maximum resistance load of the double-notched push-through specimen.

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Table 4
Points of the bilinear tension softening diagram for steel fiber-reinforced (F-15-2 mixture).

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Table 5
Mechanical properties of concrete.

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Table 6
Mechanical properties of beam reinforcements.

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Table 7
Bilinear tension softening of the steel fiber-reinforced concrete of beam V-1-0.

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Table 8
Maximum load of beams estimated by the finite element model using the smeared crack approach.

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Table 9
Maximum load of beams estimated by the finite element model using the discrete crack approach.

(1)

(2)

(3)

(4)

Mixture	v_f (%)	$f_{c m}$ (MPa)		$f_{c t m}$ (MPa)		$E_{c m}$ (GPa)		$G_{f}^{I}$ (N/mm)
Mixture	v_f (%)	Mean	SD	Mean	SD	Mean	SD	Mean	SD
F-15-2	1.50	68.74	8.57	3.93	1.02	40.39	1.84	5.96	0.97
F-00-2	0.00	65.08	4.73	4.13	0.90	39.04	0.74	0.16	0.01
F-00-1	0.00	66.29	6.54	5.18	-	37.96	1.67	-	-

Stage	Shear stress (τ) (MPa)	Crack width (w) (mm)	α (MPa/mm)	β	Normal strain (ε_nn) x10^-3
1	2.0	0.008	250	1	0.107
2	2.4	0.015	57	0.228	0.200
3	-	-	-	0.01	4.500

Shear retention factor	Maximum load (kN)	Difference from test result
Experimental result (F-00-2)	79.62 ± 3.2	-
β = 0.01 (constant)	68.71	−13.70%
β = 0.02 (constant)	71.15	−10.64%
β = 0.03 (constant)	73.28	−7.96%
β = 0.04 (constant)	76.41	−4.03%
β = 0.05 (constant)	78.62	−1.25%
Al Mahaidi [¹⁵15 DIANA FEA, User’s Manual - Material Library - DIANA Release 10.0, 1st ed. Delft: TNO DIANA BV, 2016.] (variable)	70.51	−11.44%
Červenka et al. [¹⁸18 V. Červenka, L. Jendele, and J. Červenka, ATENA Program Documentation - Part 1 - Theory. Prague: Cervenka Consulting Ltd., 2005.] (variable)	76.79	−3.55%
Rots and Blaauwendraad, k = 1 [¹⁷17 J. G. Rots and J. Blaauwendraad, "Crack models for concrete: discrete or smeared? Fixed, multi-directional or rotating," Heron, vol. 34, no. 1, pp. 1–59, 1989.] (variable)	95.04	19.37%
Rots and Blaauwendraad, k = 2 [¹⁷17 J. G. Rots and J. Blaauwendraad, "Crack models for concrete: discrete or smeared? Fixed, multi-directional or rotating," Heron, vol. 34, no. 1, pp. 1–59, 1989.] (variable)	86.17	8.23%
Figueiras [¹⁶16 J. A. Figueiras, Ultimate Load Analysis of Anisotropic and Reinforced Concrete Plates and Shells. Univ. Coll. Swansea, 1983. Accessed: Sep. 18, 2022. [Online]. Available: https://repositorio-aberto.up.pt/bitstream/10216/11772/2/Texto integral.pdf https://repositorio-aberto.up.pt/bitstre... ] (variable)	78.41	−1.52%
Proposed β factor (variable)	79.43	0.24%

Point	Tensile stress (MPa)	Total strain (x10^-3)
1	3.93	0.09
2	1.80	50
3	0.7	1000

Beam	Fiber content (%)	Coarse aggregate (mm)	$f_{c}$ (MPa)	$f_{c t}$ (MPa)	$E_{c}$ (GPa)	$G_{f}^{I}$ (N/mm)	$G_{c}$ (N/mm)	𝜈
V-0-0	0.00	12.50	46.30	3.70	27.73	0.146	14.6	0.3
V-1-0	1.00	12.50	56.87	3.28	31.78	Table 6	341.1	0.3
1R	0.00	19.00	70.20	3.24	28.82	0.156	15.59	0.3

Beam	1R			V-0-0 and V-1-0
Bar diameter (mm)	6.3	8.0	20.0	6.3	20.0
Modulus of elasticity (GPa)	173.33	203.57	162.02	210.00	210.00
Yield strength (MPa)	390	570	546	600	500

Point	Tensile stress (MPa)	Strain (‰)
1	3.28	0.103
2	1.64	12.00
3	0.00	50.00

Shear retention factor	V-0-0		V-1-0		1R
Shear retention factor	Maximum load (kN)	Difference from test	Maximum load (kN)	Difference from test	Maximum load (kN)	Difference from test
Experimental result	172.50	-	260.00	-	133.00	-
β = 0.01	145.66	−15.56%	276.77	6.45%	153.67	15.54%
β = 0.05	153.95	−10.75%	263.97	1.53%	128.75	−3.20%
β = 0.10	153.77	−10.86%	248.54	−4.41%	127.46	−4.17%
β = 0.30	184.76	7.11%	287.71	10.66%	132.88	−0.09%
Al Mahaidi [¹⁵15 DIANA FEA, User’s Manual - Material Library - DIANA Release 10.0, 1st ed. Delft: TNO DIANA BV, 2016.]	169.05	−2.00%	170.10	−34.58%	135.62	1.97%
Červenka et al. [¹⁸18 V. Červenka, L. Jendele, and J. Červenka, ATENA Program Documentation - Part 1 - Theory. Prague: Cervenka Consulting Ltd., 2005.]	198.95	15.33%	279.78	7.61%	128.07	−3.71%
Rots and Blaauwendraad, k = 1 [¹⁷17 J. G. Rots and J. Blaauwendraad, "Crack models for concrete: discrete or smeared? Fixed, multi-directional or rotating," Heron, vol. 34, no. 1, pp. 1–59, 1989.]	204.11	18.32%	231.34	11.02%	128.56	−3.34%
Figueiras [¹⁶16 J. A. Figueiras, Ultimate Load Analysis of Anisotropic and Reinforced Concrete Plates and Shells. Univ. Coll. Swansea, 1983. Accessed: Sep. 18, 2022. [Online]. Available: https://repositorio-aberto.up.pt/bitstream/10216/11772/2/Texto integral.pdf https://repositorio-aberto.up.pt/bitstre... ]	177.13	2.68%	248.54	−4.41%	105.10	−20.98%
Proposed β factor	174.47	1.14%	282.44	8.63%	138.74	4.32%

Crack dilatancy model	V-0-0		1R
Crack dilatancy model	Maximum load (kN)	Difference from test	Maximum load (kN)	Difference from test
Experimental result	172.50	-	133.00	-
Li et al. [⁹9 B. Li, K. Maekawa, and H. Okamura, "Contact density model for stress transfer across cracks in concrete," J. Fac. Eng. Univ. Tokyo, vol. XL, no. 1, pp. 9–52, 1989.]	164.05	−4.89%	141.73	6.62%
Walraven and Reinhardt [⁷7 J. C. Walraven and H. W. Reinhardt, "Theory and experiments on the mechanical behaviour of cracks in plain and reinforced concrete subjected to shear loading," Heron, vol. 26, no. 1A, pp. 1–68, 1981.]	172.09	−0.24%	140.73	5.71%
Walraven [6]	134.18	−22.21%	137.35	3.27%
Gambarova and Karakoç [⁸8 P. G. Gambarova and C. Karakoc, “A new approach to the analysis of the confinement role in regularly cracked concrete elements,” in H - Struct. Eng. Prestressed Reac. Pressure Vessels Other Struct. H5 - Failure, Fracture Propag. Ductility SMiRT 7, 1983, pp. 251–261.]	155.21	−10.03%	143.62	7.98%

[1] Corresponding author: Daniel de Lima Araújo. E-mail: dlaraujo@ufg.br

Brasil

Brasil

Computational modeling of plain and steel fiber-reinforced concrete beams without transverse reinforcement

Modelagem computacional de vigas de concreto simples e reforçados com fibras de aço sem armadura transversal

Abstract