Dimensionamento à flexão de vigas em concreto armado com barras de FRP

Barbosa, Felipe Augusto da Silva; Bittencourt, Túlio Nogueira; Boriolo, Gustavo Rodovalho; André, Fellipe Rodrigues; Futai, Marcos Massao

Felipe Augusto da Silva Barbosa

Universidade de São Paulo – USP, Department of Structural and Geotechnical Engineering, São Paulo, SP, Brasil

https://orcid.org/0000-0002-5632-283X

Túlio Nogueira Bittencourt

Universidade de São Paulo – USP, Department of Structural and Geotechnical Engineering, São Paulo, SP, Brasil

https://orcid.org/0000-0001-6523-2687

Gustavo Rodovalho Boriolo

Universidade de São Paulo – USP, Department of Structural and Geotechnical Engineering, São Paulo, SP, Brasil

https://orcid.org/0000-0001-5201-2191

Fellipe Rodrigues André

Universidade São Judas Tadeu – USJT, Department of Civil Engineering, São Paulo, SP, Brasil

https://orcid.org/0000-0002-0192-0584

Marcos Massao Futai

Universidade de São Paulo – USP, Department of Structural and Geotechnical Engineering, São Paulo, SP, Brasil

https://orcid.org/0000-0002-4969-3085

Corresponding author: Felipe Augusto da Silva Barbosa. E-mail: felipe.asb92@usp.br

Conflict of interest: Nothing to declare.

Author contributions: FASB: conceptualization, writing, data curation, formal analysis and methodology; TNB and MMF: conceptualization, funding acquisition, supervision; FRA: conceptualization and methodology; GRB: conceptualization

Editors: Rebecca Gravina, Guilherme Aris Parsekian.

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Figure 1
Concrete constitutive model for compression, for which 0.85f_cd and f_ck correspond to the design and characteristic strength, respectively [⁹9 Associação Brasileira de Normas Técnicas, Projeto de Estruturas de Concreto, ABNT NBR 6118, 3ª ed., 2014, 238 p.]

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Figure 2
Equilibrium and compatibility of a balanced cross-section with the simplified stress block located on the flange [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.]

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Figure 3
Compatibility and equilibrium of a balanced cross-section with the simplified stress block reaching the web [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.]

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Figure 4
Equilibrium and compatibility conditions of a tension-controlled section with neutral axis on the flange and concrete exhibiting linear behavior [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 5
Equilibrium and compatibility conditions of a tension-controlled section for which concrete behaves non-linearly [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 6
Compatibility and equilibrium of a tension-controlled cross-section for which the concrete linear stress-strain relationship applies [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 7
Compatibility and equilibrium of a tension-controlled section for which the concrete behaves non-linearly [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 8
Equilibrium and compatibility of a compression-controlled section with stress block on the flange [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 9
Equilibrium and compatibility of a compression-controlled section whose stress block reaches the web [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 4
Equilibrium and compatibility conditions of a tension-controlled section with neutral axis on the flange and concrete exhibiting linear behavior [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 5
Equilibrium and compatibility conditions of a tension-controlled section for which concrete behaves non-linearly [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 6
Compatibility and equilibrium of a tension-controlled cross-section for which the concrete linear stress-strain relationship applies [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 7
Compatibility and equilibrium of a tension-controlled section for which the concrete behaves non-linearly [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 8
Equilibrium and compatibility of a compression-controlled section with stress block on the flange [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 9
Equilibrium and compatibility of a compression-controlled section whose stress block reaches the web [⁸8 F. A. S. Barbosa, “Numerical assessment of concrete flexural members reinforced by FRP rebars,” M.S. thesis, Univ. São Paulo, São Paulo, 2020.].

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Figure 10
Compatibility and equilibrium to compute the sustained stress considering the almost permanent load combination [¹²12 R. C. Carvalho and J. R. Figueiredo Fo., Cálculo e Detalhamento de Estruturas Usuais de Concreto Armado: Segundo a NBR 6118:2014, 4ª ed. São Carlos: Edufscar, 2014, 415 p.].

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Figure 11
– Correlation scatter plot for the experimental M_exp. and theoretical M_th flexural capacities

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Figure 12
Computation of the AFRP reinforcement area using the developed design program.

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Table 1
- Comparison between experimental and predicted flexural capacities

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Table 2
Definition of the failure mode corresponding to all design approaches previously described

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Table 3
Design for the ultimate limit state and checking for creep rupture for all approaches

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Ref.		Beam		Geometrical features						Materials (MPa)						Ultimate Moments (kNm)				Failure
Ref.		Beam		b (cm)		h(cm)		ρ_f (%)		f'_c		f_fu		10^-3 E_f		M_exp		M_th	M_th/M_exp	Failure
[¹³13 C. Kassem, A. S. Farghaly, and B. Benmokrane, "Evaluation of flexural behavior and serviceability performance of concrete beams reinforced with FRP bars," J. Compos. Constr., vol. 5, no. 11, pp. 682–695, 2011.]		C1-4		20		30		0.47		40.4		1368.0		114.0		71.20		71.72	1.01	TC
		C1-4b		20		30		0.47		40.4		1368.0		114.0		74.58		80.20	1.08	CC
		C1-6		20		30		0.71		39.3		1368.0		114.0		83.13		83.53	1.00	CC
		C1-8		20		30		0.95		39.3		1368.0		114.0		90.39		92.43	1.02	CC
		C2-4		20		30		0.43		39.9		1904.0		112.0		78.75		69.38	0.88	CC
		C2-4b		20		30		0.43		39.9		1904.0		112.0		78.18		76.16	0.97	CC
		C2-6		20		30		0.64		40.8		1904.0		112.0		80.89		82.09	1.01	CC
		C2-8		20		30		0.85		40.8		1904.0		112.0		89.39		91.09	1.02	CC
		G1-6		20		30		1.29		39.1		600.0		40.0		77.47		68.86	0.89	CC
		G1-8		20		30		1.72		39.1		600.0		40.0		86.76		76.77	0.88	CC
		G2-6		20		30		1.13		39.1		648.0		36.0		71.00		63.03	0.89	CC
		G2-8		20		30		1.51		39.1		648.0		36.0		84.54		70.52	0.83	CC
		AR-6		20		30		0.71		39.1		1716.0		52.0		70.85		61.62	0.87	CC
		AR-8		20		30		0.95		39.1		1716.0		52.0		71.75		69.04	0.96	CC
[¹⁴14 M. Thériault and B. Benmokrane, "Effects of FRP reinforcement ratio and concrete strength on flexural behavior of concrete beams," J. Compos. Constr., vol. 2, no. 1, pp. 7–16, 1998.]		BC4NA		13		18		2.03		46.2		772.9		38.0		22.40		20.13	0.90	CC
		BC4NB		13		18		2.03		46.2		772.9		38.0		20.60		20.13	0.98	CC
		BC4HA		13		18		2.03		53.9		772.9		38.0		21.00		20.63	0.98	CC
		BC4HB		13		18		2.03		53.9		772.9		38.0		21.40		20.63	0.96	CC
		BC2NA		13		18		1.02		53.1		772.9		38.0		21.90		18.89	0.86	CC
		BC2NB		13		18		1.02		53.1		772.9		38.0		20.00		18.89	0.94	CC
		BC2HA		13		18		1.02		57.2		772.9		38.0		19.70		18.48	0.94	CC
		BC4HB		13		18		1.02		57.2		772.9		38.0		20.60		18.48	0.90	CC
[¹⁵15 R. Masmoudi, M. Thériault, and B. Benmokrane, "Flexural behavior of concrete beams reinforced with deformed fiber reinforced plastic reinforcing rods," ACI Struct. J., vol. 5, no. 6, pp. 665–675, 1998.]		CB2B-1		20		30		0.58		52.0		779.0		38.0		57.90		65.04	1.12	TC
		CB2B-2		20		30		0.58		52.0		779.0		38.0		59.80		65.04	1.09	TC
		CB3B-1		20		30		0.87		52.0		779.0		38.0		66.00		79.30	1.20	CC
		CB3B-2		20		30		0.87		52.0		779.0		38.0		64.80		79.30	1.22	CC
		CB4B-1		20		30		1.16		45.0		779.0		38.0		75.40		74.63	0.99	CC
		CB4B-2		20		30		1.16		45.0		779.0		38.0		71.70		74.63	1.04	CC
		CB6B-1		20		30		1.74		45.0		779.0		38.0		84.80		87.48	1.03	CC
		CB6B-2		20		30		1.74		45.0		779.0		38.0		85.40		87.48	1.02	CC
[¹⁶16 Ridwan and D. T. Putri, "Flexural capacity of concrete beam reinforced with GFRP bars," J. Phys. Conf. Ser., vol. 2049, no. 1, pp. 012075, 2021.]		FB-2		30		40		0.22		30.0		690.0		41.0		68.94		62.75	0.91	TC
		FB-3		30		40		0.33		30.0		690.0		41.0		111.18		93.04	0.84	TC
		FB-4		30		40		0.44		30.0		690.0		41.0		125.88		121.91	0.97	TC
		FB-6		30		40		0.66		30.0		690.0		41.0		171.54		158.56	0.92	CC
		FB-8		30		40		0.88		30.0		690.0		41.0		222.60		178.05	0.80	CC
		HFB-3		30		40		0.33		50.0		690.0		41.0		93.24		94.89	1.02	TC
		HFB-4		30		40		0.44		50.0		690.0		41.0		119.04		124.63	1.05	TC
		HFB-6		30		40		0.66		60.0		690.2		41.0		200.46		185.69	0.93	TC
		HFB-8		30		40		0.88		50.0		690.0		41.0		218.04		241.47	1.11	TC
		HFB-10		30		40		1.11		50.0		690.0		41.0		219.36		265.55	1.21	CC
[¹⁷17 B. Benmokrane, O. Chaallal, and R. Masmoudi, "Glass Fibre Reinforced Plastic (GFRP) rebars for concrete structures," Constr. Build. Mater., vol. 9, no. 6, pp. 353–364, 1995.]^* * Direct tensile tests not performed by the authors, FRP properties provided by the manufacturer, TC = Tension-controlled; CC = Compression-controlled		ISO30-2		20		30		0.95		44.0		688.8		42.0		80.40		85.18	1.06	CC
		KD30-1		20		30		0.95		44.0		640.9		49.0		50.60		79.79	1.58	CC
		KD30-2		20		30		0.95		44.0		640.9		49.0		63.80		79.79	1.25	CC
		KD45-1		20		45		0.64		55.0		640.9		49.0		106.60		136.16	1.28	TC
		KD45-2		20		45		0.64		55.0		640.9		49.0		113.00		136.16	1.20	TC
		ISO55-1		20		55		0.52		43.0		688.8		42.0		181.50		184.17	1.01	TC
		ISO55-2		20		55		0.52		43.0		688.8		42.0		181.50		184.17	1.01	TC
		KD55-1		20		55		0.52		43.0		640.9		49.0		146.90		171.48	1.17	TC
		KD55-2		20		55		0.52		43.0		640.9		49.0		172.50		171.48	0.99	TC
[¹⁸18 B. Benmokrane, O. Chaallal, and R. Masmoudi, "Flexural response of concrete beams reinforced with FRP reinforcing bars," ACI Struct. J., vol. 91, no. 2, pp. 46–55, 1996.]*		ISO2		20		30		0.95		43.0		688.5		45.0		80.40		83.26	1.04	CC
		ISO3		20		55		0.52		43.0		688.5		45.0		181.70		189.95	1.05	TC
		ISO4		20		55		0.52		43.0		688.5		45.0		181.70		189.95	1.05	TC
[¹⁹19 A. F. Ashour, "Flexural and shear capacities of concrete beams reinforced with GFRP bars," Constr. Build. Mater., vol. 20, pp. 1005–1015, 2006.]		Beam 2		15		20		0.19		27.7		650.0		38.0		5.89		5.84	0.99	TC
		Beam 4		15		25		0.15		27.7		650.0		38.0		7.85		7.94	1.01	TC
		Beam 6		15		30		0.13		27.7		650.0		38.0		10.79		9.77	0.91	TC
		Beam 8		15		20		0.19		50.1		650.0		38.0		5.89		5.90	1.00	TC
		Beam 10		15		25		0.15		50.1		650.0		38.0		9.48		8.00	0.84	TC
		Beam 12		15		30		0.25		50.1		650.0		38.0		16.75		19.46	1.16	TC
[²⁰20 J. R. Yost, C. H. Goodspeed, and E. R. Schmeckpeper, "Flexural performance of concrete beams reinforced with FRP grids," J. Compos. Constr., vol. 5, no. 1, pp. 18–25, 2005.]*		1FRP1		38 .1		20.3		0.10		27.6		829.7		41.4		11.49		11.60	1.01	TC
		1FRP2		38.1		20.3		0.10		27.6		829.7		41.4		12.67		11.60	0.92	TC
		1FRP3		38.1		20.3		0.10		27.6		829.7		41.4		11.49		11.60	1.01	TC
		2FRP1		31.8		21.6		0.12		27.6		829.7		41.4		13.62		12.46	0.91	TC
		2FRP2		31.8		21.6		0.12		27.6		829.7		41.4		13.26		12.46	0.94	TC
		2FRP3		31.8		21.6		0.12		27.6		829.7		41.4		13.06		12.46	0.95	TC
		4FRP1		20.3		15.2		1.04		27.6		829.7		41.4		15.78		14.68	0.93	TC
		4FRP2		20.3		15.2		1.04		27.6		829.7		41.4		15.58		14.68	0.94	TC
		4FRP3		20.3		15.2		1.04		27.6		829.7		41.4		16.29		14.68	0.90	TC
		5FRP1		19.1		15.2		1.10		27.6		829.7		41.4		16.37		14.14	0.86	CC
		5FRP2		19.1		15.2		1.10		27.6		829.7		41.4		16.65		14.14	0.85	CC
		5FRP3		19.1		15.2		1.10		27.6		829.7		41.4		15.78		14.14	0.90	CC
[²¹21 A. F. Ashour and M. N. Habeeb, "Continuous concrete beams reinforced with CFRP bars," ICE Proc. Struct. Build., vol. 161, no. 6, pp. 349–357, 2008.]		C-S-1		20		30		0.75		26.9		1000.0		200.0		64.11		54.50	0.85	TC
		C-S-2		20		30		0.29		27.5		2000.0		200.0		44.28		43.68	0.99	TC
		C-C-3		20		30		0.52		23.6		2000.0		200.0		44.76		43.30	0.97	TC
		C-C-4		20		30		0.52		27.2		1000.0		200.0		60.66		54.87	0.90	TC
		C-C-5		20		30		0.75		28.0		1000.0		200.0		56.03		54.64	0.98	TC
[²²22 A. F. Ashour, and M. Family, "Tests of concrete flanged beams reinforced with CFRP bars," Mag. Concr. Res., vol. 58, no. 9, pp. 627–639, 2006.]		T/C150-2		20/70		70/15		0.36		37.5		1060.0		200.0		64.89		76.77	1.18	TC
		T/C150-4		20/70		70/15		0.52		37.5		1060.0		200.0		145.69		146.89	1.01	TC
		T/C100-4		20/70		70/10		0.63		37.5		1060.0		200.0		104.63		123.23	1.18	TC
		R/C-2		20		35		0.47		40.5		1060.0		200.0		57.33		71.05	1.24	TC
		R/C-4		20		35		0.79		40.5		1060.0		200.0		124.60		136.60	1.10	TC
[²³23 S. H. Alsayed, Y. A. Al-Salloum, and T. H. Almusallam, "Performance of glass fiber reinforced plastic bars as a reinforcing material for concrete structures," Compos., Part B Eng., vol. 31, pp. 555–567, 2000.]		II		20		21		2.70		31.3		700.0		35.6		34.19		36.84	1.08	CC
		III		20		26		0.98		31.3		886.0		43.4		45.13		47.67	1.06	CC
		IV		20		30		0.95		40.7		700.0		35.6		59.19		69.84	1.18	CC
		V		20		25		2.27		40.7		700.0		35.6		57.00		63.54	1.11	CC
[²⁴24 C. Barris, L. Torres, A. Turom, M. Baena, and A. Catalan, "An experimental study of the flexural behavior of GFRP RC beams and comparison with predicted models," Compos. Struct., vol. 91, pp. 286–295, 2009.]		C-212-D1		14		19		0.85		59.8		1353.0		63.3		38.20		27.13	0.71	CC
		C-216-D1		14		19		1.51		56.3		995.0		64.2		45.06		34.09	0.76	CC
		C-316-D1		14		19		2.27		55.2		995.0		64.2		49.38		39.75	0.80	CC
		C-212-D2		16		19		0.74		39.6		1353.0		63.3		27.69		21.64	0.78	CC
		C-216-D2		16		19		1.32		61.7		995.0		64.2		42.15		29.17	0.69	CC
		C-316-D2		16		19		1.98		60.1		995.0		64.2		43.20		34.07	0.79	CC
[²⁵25 M. El-Mogy, A. El-Ragaby, and E. El-Salakawy, "Flexural behavior of continuous FRP-reinforced concrete beams," J. Compos. Constr., vol. 14, no. 6, pp. 669–680, 2010.]		GS1		20		30		1.65		28.0		736.0		46.0		60.20		62.59	1.04	CC
		GS2		20		30		1.65		26.0		736.0		46.0		49.00		50.70	1.03	CC
		CS1		20		30		0.59		26.0		1392.0		116.0		51.80		58.99	1.14	CC
[²⁶26 G. Thiagarajan, "Experimental and Analytical Behavior of Carbon Fiber-based rods as Flexural Reinforcement," J. Compos. Constr., vol. 7, no. 1, pp. 64–72, 2003.]*		B4		15.24		15.24		0.27		51.7		1899.9		140.0		12.60		13.84	1.10	TC
		B5		15.24		15.24		0.27		48.0		1899.9		140.0		10.15		13.76	1.36	TC
		B6		15.24		15.24		0.27		45.9		1899.9		140.0		12.87		13.73	1.07	TC
		B7		15.24		15.24		0.43		49.3		1899.9		140.0		17.10		18.49	1.08	CC
		B8		15.24		15.24		0.43		51.1		1899.9		140.0		16.92		18.45	1.09	CC
		B9		15.24		15.24		0.43		53.3		1899.9		140.0		16.58		18.21	1.10	CC
		B10		15.24		15.24		0.61		53.4		1899.9		140.0		17.85		20.74	1.16	CC
		B11		15.24		15.24		0.61		55.0		1899.9		140.0		17.61		20.59	1.17	CC
		B12		15.24		15.24		0.61		43.9		1899.9		140.0		17.51		19.52	1.12	CC
[²⁷27 M. A. Rashid, M. A. Mansur, and P. Paramasivam, "Behavior of aramid fiber-reinforced polymer reinforced high strength concrete beams under bending," J. Compos. Constr., vol. 9, no. 2, pp. 117–127, 2005.]		AF2T1		15		30		0.35		42.8		1759.6		53.0		44.17		46.58	1.05	CC
		BF3T1		15		30		0.52		85.8		1759.6		53.0		59.46		60.81	1.02	CC
		CF3T1		15		30		0.52		85.6		1759.6		53.0		67.21		60.75	0.90	CC
		DF2T1		15		30		0.35		84.5		1759.6		53.0		48.06		50.28	1.05	CC
		DF3T1		15		30		0.52		84.5		1759.6		53.0		62.77		60.40	0.96	CC
		DF4T1		15		30		0.70		84.5		1759.6		53.0		60.02		62.06	1.03	CC
		DF3T2		15		30		0.52		84.5		1759.6		53.0		62.41		60.40	0.97	CC
		DF3T3		15		30		0.52		84.5		1759.6		53.0		60.80		60.40	0.99	CC
		DS4T2		15		30		1.79		84.5		1759.6		53.0		107.20		89.01	0.83	CC
[²⁸28 R. K. Kalluri, “Bending behavior of concrete T-beams reinforced with glass-fiber reinforced polymer bars,” Ph.D. dissertation, West Virginia Univ., Morgantown, 1999, 112 p.]*		B1T1		15/91		46/7.6		1.07		38.6		620.0		40.0		111.85		106.63	0.95	TC
		B2T1		15/76		46/7.6		1.17		35.2		620.0		40.0		107.79		106.58	0.99	TC
		B3C1		15/46		45/6.4		4.31		21.4		620.0		40.0		203.37		208.13	1.02	CC
		B4T1		15/107		45/6.4		1.27		18.6		620.0		40.0		156.60		144.44	0.92	TC
		B3C2		15/61		37/6.4		3.73		21.4		620.0		40.0		223.71		198.51	0.89	CC
		B1T2		15/91		38/7.6		1.83		38.6		620.0		40.0		213.54		248.59	1.16	TC
		B2C1		15/76		38/7.6		2.57		35.2		620.0		40.0		244.05		297.66	1.22	CC
		B4C1		15/107		37/6.4		3.31		18.6		620.0		40.0		244.05		275.70	1.13	CC
[²⁹29 H. A. Toutanji and M. Saafi, "Flexural behavior of concrete beams reinforced with glass fiber-reinforced polymer (GFRP) bars," ACI Struct. J., vol. 97, no. 5, pp. 712–719, 2000.]	GB1-1		18		30		0.47		35.0		695.0		40.0		60.00		44.82		0.75	TC
	GB1-2		18		30		0.47		35.0		695.0		40.0		59.00		44.82		0.76	TC
	GB2-1		18		30		0.71		35.0		695.0		40.0		65.00		60.69		0.93	CC
	GB2-2		18		30		0.71		35.0		695.0		40.0		64.30		60.69		0.94	CC
	GB3-1		18		30		0.94		35.0		695.0		40.0		71.00		63.27		0.89	CC
	GB3-2		18		30		0.94		35.0		695.0		40.0		70.50		63.27		0.90	CC

Case	FRP	f_fu* (MPa)	Exposition	E_f (GPa)	f_ck (MPa)	M_d (kNm)	x_b (cm)	M_b (kNm)	Failure Mode
1A	Aramid	2540	No	125	80	2100	14.1	5276.5	TC
1B	Carbon	3690	No	150	60	1820	11.9	4088.2	TC
2A	Glass	683	Yes	51	65	962.7	24.8	5144.1	TC
2B	Glass	600	No	51	40	759.4	29.3	3575.6	TC
2C	Carbon	1300	Yes	145	90	1820	26.6	6142.7	TC
2D	Aramid	2300	Yes	100	20	1582	17.8	1629.0	TC
3A	Carbon	3000	No	120	80	4340	10.7	4149.0	CC
3B	Carbon	3690	Yes	180	90	5670	13.9	5444.6	CC
4A	Aramid	1800	Yes	70	20	1820	16.3	1606.2	CC

Case	x (cm)	A_f (cm²)	M_apc (kNm)	f_fs (MPa)	A_adj (cm²)	A_b (cm²)	Failure	x_adj (cm)	ε_c (‰)	ε_f (‰)	M_r (kNm)	10³φ (rad/m)
1A	5.3	13.0	1010	841.4	16.7	35.3	TC	6.8	1.10	14.1	2678.3	0.17
1B	5.1	7.3	810	1244	7.3	16.9	TC	5.1	1.15	18.9	1820.0	0.22
2A	8.6	30.0	325	122	38.7	166.5	TC	9.7	0.87	7.21	1233.4	0.09
2B	2.9	23.2	360	175.5	43	116.4	TC	5.5	0.47	7.24	1394.4	0.09
2C	11.6	23.5	880	431	23.5	81.2	TC	11.6	0.92	6.21	1820.0	0.08
2D	14.7	13.2	549	477	13.2	13.6	TC	14.7	2.76	14.15	1582.0	0.19
3A	11.3	23.1	1700	841.9	23.1	20.9	CC	11.3	2.60	18.22	4340.0	0.23
3B	16.3	31.5	3350	1307.4	31.5	25.0	CC	16.3	2.6	11.78	5670.0	0.16
4A	31.9	49.2	1090	279	49.2	17.1	CC	31.9	3.5	6.39	1820.0	0.11

{\bar{y}}_{lin} = \frac{\frac{b_{w} {x_{lin}}^{2}}{3} + {h_{f}}^{2} (1 - \frac{2}{3} \frac{h_{f}}{x_{lin}}) (b_{f} - b_{w})}{b_{w} x_{lin} + h_{f} (2 - \frac{h_{f}}{x_{lin}}) (b_{f} - b_{w})}

I_{cr} = \frac{(b_{f} - b_{w}) {h_{f}}^{3}}{12} + \frac{b_{w} x_{cr}^{3}}{3} + (b_{f} - b_{w}) {(x_{cr} - \frac{h_{f}}{2})}^{2} + η_{f} A_{f} {(x_{cr} - d)}^{2}

[1] Corresponding author: Felipe Augusto da Silva Barbosa. E-mail: felipe.asb92@usp.br

Brasil

Brasil

Dimensionamento à flexão de vigas em concreto armado com barras de FRP

Resumo