Simulação numérica de vigas mistas de aço e concreto: estratégias atuais de modelagem por elementos finitos

Benincá, Matheus Erpen; Schmitz, Rebeca Jéssica; Morsch, Inácio Benvegnu

Matheus Erpen Benincá

Universidade Federal do Rio Grande do Sul – UFRGS, Escola de Engenharia, Programa de Pós-graduação em Engenharia Civil, Porto Alegre, RS, Brasil

https://orcid.org/0000-0003-4852-6127

Rebeca Jéssica Schmitz

Universidade Federal do Rio Grande do Sul – UFRGS, Escola de Engenharia, Programa de Pós-graduação em Engenharia Civil, Porto Alegre, RS, Brasil

https://orcid.org/0000-0001-5113-8972

Inácio Benvegnu Morsch

Universidade Federal do Rio Grande do Sul – UFRGS, Escola de Engenharia, Programa de Pós-graduação em Engenharia Civil, Porto Alegre, RS, Brasil

https://orcid.org/0000-0002-2473-3474

Corresponding author: Matheus Erpen Benincá. E-mail: matheuseb@hotmail.com

Conflict of interest: Nothing to declar

Author contributions: Conceptualization, ME Benincá, RJ Schmitz and IB Morsch; Investigation, ME Benincá and RJ Schmitz; Methodology, ME Benincá, RJ Schmitz and IB Morsch; Software, ME Benincá and RJ Schmitz; Supervision, IB Morsch; Validation, ME Benincá, RJ Schmitz and IB Morsch; Visualization, ME Benincá; Writing - original draft, ME Benincá; Writing - review & editing, ME Benincá, RJ Schmitz and IB Morsch.

Editors: José Marcio Calixto, José Luiz Antunes de Oliveira e Sousa, Guilherme Aris Parsekian

Model	Software	Steel profile	Connectors	Concrete slab
Gattesco [¹²12 N. Gattesco, "Analytical modeling of nonlinear behavior of composite beams with deformable connection," J. Construct. Steel Res., vol. 52, no. 2, pp. 195–218, 1999, http://dx.doi.org/10.1016/S0143-974X(99)00026-7. http://dx.doi.org/10.1016/S0143-974X(99)... ]	Authorial	Bar elements. Nonlinear hardening.	Spring elements. Curve from push-out test.	Bar elements. Nonlinear constitutive model.
CEMACOM [¹³13 J. L. P. Tamayo, I. B. Morsch, and A. M. Awruch, "Short-time numerical analysis of steel–concrete composite beams," J. Braz. Soc. Mech. Sci. Eng., vol. 37, no. 4, pp. 1097–1109, 2015, http://dx.doi.org/10.1007/s40430-014-0237-9. http://dx.doi.org/10.1007/s40430-014-023... ]–[¹⁵15 M. M. Dias, “Análise numérica de vigas mistas aço-concreto pelo método dos elementos finitos: efeitos de longa duração,” M.S. thesis, Prog. Pós-grad. Eng. Civ., Univ. Fed. Rio Grande do Sul, Porto Alegre, 2013. [Online]. Available: https://hdl.handle.net/10183/77726 https://hdl.handle.net/10183/77726... ]	Authorial	Shell elements. von Mises criterion with linear hardening.	Bar elements. Curve from push-out test.	Degenerated shell elements. Modified Drucker-Prager model.
Kotinda [¹1 T. I. Kotinda, “Modelagem numérica de vigas mistas aço-concreto simplesmente apoiadas: ênfase ao estudo interface laje-viga,” M.S. thesis, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2006. http://dx.doi.org/10.11606/D.18.2006.tde-22062006-110301. http://dx.doi.org/10.11606/D.18.2006.tde... ]	ANSYS	Shell elements SHELL43. von Mises criterion with nonlinear hardening [¹²12 N. Gattesco, "Analytical modeling of nonlinear behavior of composite beams with deformable connection," J. Construct. Steel Res., vol. 52, no. 2, pp. 195–218, 1999, http://dx.doi.org/10.1016/S0143-974X(99)00026-7. http://dx.doi.org/10.1016/S0143-974X(99)... ].	Bar elements BEAM189. von Mises criterion with linear hardening.	Hexahedral elements SOLID65. Model concrete [¹⁷17 K. J. Willam and E. P. Warnke, “Constitutive model for the triaxial behavior of concrete,” in IABSE Seminar on Concrete Structures Subjected to Triaxial Stresses, Bergamo, Italy, 1974.].
Queirozet al. [⁸8 F. D. Queiroz, P. C. G. S. Vellasco, and D. A. Nethercot, "Finite element modelling of composite beams with full and partial shear connection," J. Construct. Steel Res., vol. 63, no. 4, pp. 505–521, 2007, http://dx.doi.org/10.1016/j.jcsr.2006.06.003. http://dx.doi.org/10.1016/j.jcsr.2006.06... ]	ANSYS	Shell elements SHELL43. von Mises criterion with nonlinear hardening [¹²12 N. Gattesco, "Analytical modeling of nonlinear behavior of composite beams with deformable connection," J. Construct. Steel Res., vol. 52, no. 2, pp. 195–218, 1999, http://dx.doi.org/10.1016/S0143-974X(99)00026-7. http://dx.doi.org/10.1016/S0143-974X(99)... ].	Spring elements COMBIN39. Curve from push-out test.	Hexahedral elements SOLID65. Model concrete [¹⁷17 K. J. Willam and E. P. Warnke, “Constitutive model for the triaxial behavior of concrete,” in IABSE Seminar on Concrete Structures Subjected to Triaxial Stresses, Bergamo, Italy, 1974.].
Present analysis	ANSYS	Shell elements SHELL181. von Mises criterion with nonlinear hardening [¹²12 N. Gattesco, "Analytical modeling of nonlinear behavior of composite beams with deformable connection," J. Construct. Steel Res., vol. 52, no. 2, pp. 195–218, 1999, http://dx.doi.org/10.1016/S0143-974X(99)00026-7. http://dx.doi.org/10.1016/S0143-974X(99)... ].	Spring elements COMBIN39. Curve from push-out test.	Hexahedral elements SOLID186. Models: (i) Usermat (ii) DP-Concrete

	Beam		A2	E1	Unit	Source
Connectors	Total number of connectors		76	100	-	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Number of rows		2	2	-	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Height		101.6	50.8	mm	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Diameter		19	12.7	mm	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Ultimate shear force (Q_u)		120.6	63.76	kN	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Parameter m		1.2	1.15	1/mm	Ollgaardet al. [²³23 J. G. Ollgaard, R. G. Slutter, and J. W. Fisher, "Shear strength of stud connectors in lightweight and normalweight concrete," Eng. J., vol. 8, pp. 55–64, 1971. [Online]. Available: https://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/2010 https://preserve.lehigh.edu/engr-civil-e... ]
	Parameter n		0.55	0.5	-	Ollgaardet al. [²³23 J. G. Ollgaard, R. G. Slutter, and J. W. Fisher, "Shear strength of stud connectors in lightweight and normalweight concrete," Eng. J., vol. 8, pp. 55–64, 1971. [Online]. Available: https://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/2010 https://preserve.lehigh.edu/engr-civil-e... ]
Steel profile	Modulus of elasticity (E)	Flanges	20216	20520	kN/cm²	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Modulus of elasticity (E)	Web	20368	19912	kN/cm²	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Yield strength (f_y)	Flanges	23	24,5	kN/cm²	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Yield strength (f_y)	Web	23.9	29.2	kN/cm²	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Ultimate strength (f_u)	Flanges	41	45.8	kN/cm²	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Ultimate strength (f_u)	Web	41.8	45.3	kN/cm²	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Ratio ε_h/ε_y	Flanges	4.9	2.2	-	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Ratio ε_h/ε_y	Web	7.6	1	-	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Tangent modulus (E_h)	Flanges	350	350	kN/cm²	Kotinda [¹1 T. I. Kotinda, “Modelagem numérica de vigas mistas aço-concreto simplesmente apoiadas: ênfase ao estudo interface laje-viga,” M.S. thesis, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2006. http://dx.doi.org/10.11606/D.18.2006.tde-22062006-110301. http://dx.doi.org/10.11606/D.18.2006.tde... ]
	Tangent modulus (E_h)	Web	350	350	kN/cm²	Kotinda [¹1 T. I. Kotinda, “Modelagem numérica de vigas mistas aço-concreto simplesmente apoiadas: ênfase ao estudo interface laje-viga,” M.S. thesis, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2006. http://dx.doi.org/10.11606/D.18.2006.tde-22062006-110301. http://dx.doi.org/10.11606/D.18.2006.tde... ]
Concrete slab	Uniaxial compressive strength (f_cm)		2.69	3.27	kN/cm²	Chapman and Balakrishnan [⁹9 J. C. Chapman and S. Balakrishnan, "Experiments on composite beams," Struct. Eng., vol. 42, no. 11, pp. 369–383, 1964.]
	Uniaxial tensile strength (f_ctm)		0.21	0.25	kN/cm²	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Biaxial compressive strength (f_c2m)		3.16	3.82	kN/cm²	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Stress σ_com (on the compressive meridian)		-24	-24	kN/cm²	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Stress τ_com (on the compressive meridian)		15.3	15.1	kN/cm²	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Initial tangent modulus of elasticity (E_ci)		2990	3191	kN/cm²	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Reduced modulus of elasticity (E_c)		2575	2790	kN/cm²	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Secant modulus of elasticity (E_c1)		1300	1480	kN/cm²	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Strain at the max. compressive stress (ε_c1)		2.1	2.2	‰	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
	Ultimate compressive strain (ε_c,lim)		3.5	3.5	‰	FIB [²⁶26 Fédération Internationale du Béton, Fib Model Code for Concrete Structures 2010, 2013.]
Slab Reinforcement	Modulus of elasticity (E)		20500	20500	kN/cm²	Kotinda [¹1 T. I. Kotinda, “Modelagem numérica de vigas mistas aço-concreto simplesmente apoiadas: ênfase ao estudo interface laje-viga,” M.S. thesis, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2006. http://dx.doi.org/10.11606/D.18.2006.tde-22062006-110301. http://dx.doi.org/10.11606/D.18.2006.tde... ]
Slab Reinforcement	Yield strength (f_y)		32	32	kN/cm²	Kotinda [¹1 T. I. Kotinda, “Modelagem numérica de vigas mistas aço-concreto simplesmente apoiadas: ênfase ao estudo interface laje-viga,” M.S. thesis, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2006. http://dx.doi.org/10.11606/D.18.2006.tde-22062006-110301. http://dx.doi.org/10.11606/D.18.2006.tde... ]

Model		Beam A2		Beam E1
Model		Load P [kN]	(P-P_exp)/P_exp	Load P [kN]	(P-P_exp)/P_exp
Experimental	P_exp	441	-	510	-
Numerical (Usermat)	P₁	382	-13.38%	450	-11.76%
Numerical (Usermat)	P₂	458	3.85%	518	1.57%
Numerical (DP-Concrete)	P₁	363	-17.69%	403	-20.98%
Numerical (DP-Concrete)	P₂	447	1.36%	549	7.65%

λ = \{\begin{matrix} c_{1} \cdot \cos (\frac{1}{3} \arccos (c_{2} \cdot \cos (3 θ))), i f \cos (3 θ) \geq 0 \\ c_{1} \cdot \cos (\frac{π}{3} - \frac{1}{3} \arccos (- c_{2} \cdot \cos (3 θ))), i f \cos (3 θ) \leq 0 \end{matrix}

Ω_{c} = \{\begin{matrix} Ω_{c i} + (1 - Ω_{c i}) \cdot \sqrt{2 \cdot \frac{κ}{κ_{c m}} - \frac{κ^{2}}{κ_{c m}^{2}}}, i f 0 < κ \leq κ_{c m} \\ \begin{matrix} 1 - \frac{1 - Ω_{c r}}{κ_{c r} - κ_{c m}} \cdot (κ - κ_{c m}), i f κ_{c m} < κ \leq κ_{c r} \end{matrix} \\ Ω_{c r}, i f κ \geq κ_{c r} \end{matrix}

Ω_{t} = \{\begin{matrix} 1 - (1 - Ω_{t r}) \cdot \frac{κ}{κ_{t r}}, i f 0 < κ < κ_{t r} \\ Ω_{t r}, i f κ \geq κ_{t r} \end{matrix}

[1] Corresponding author: Matheus Erpen Benincá. E-mail: matheuseb@hotmail.com

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Simulação numérica de vigas mistas de aço e concreto: estratégias atuais de modelagem por elementos finitos

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