Nonlinear analysis method of concrete structures under cyclic loading based on the generalized secant modulus

Pereira, Lívia Ramos Santos; Penna, Samuel Silva

doi:10.1590/S1983-41952022000400006

Lívia Ramos Santos Pereira

Universidade Federal de Minas Gerais – UFMG, Departamento de Engenharia de Estruturas, Belo Horizonte, MG, Brasil

http://orcid.org/0000-0003-4575-6875

Samuel Silva Penna

Universidade Federal de Minas Gerais – UFMG, Departamento de Engenharia de Estruturas, Belo Horizonte, MG, Brasil

http://orcid.org/0000-0001-7256-1518

Corresponding author: Lívia Ramos Santos Pereira. E-mail: lrsp@ufmg.br

Conflict of interest: Nothing to declare.

Author contributions: LRSP, SSP: computer programming, conceptualization, methodology, writing, formal analysis.

Editors: Osvaldo Manzoli, Guilherme Aris Parsekian.

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Figure 1
Cyclic analysis process.

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Figure 2
Secant unloading/reloading.

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Figure 3
Linear elastic unloading/reloading.

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Figure 4
Linear unloading/reloading based on the focal point.

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Figure 5
Cycle control - steps count.

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Figure 6
Cycle control - load limit.

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Figure 7
Standard Newton-Raphson algorithm, including the cyclic monitoring.

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Figure 8
Direct tension simulation - geometry and mesh.

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Figure 9
Direct tension simulations: different strategies to represent the load cycles.

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Figure 10
Direct compression simulation - geometry and mesh.

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Figure 11
Direct compression simulations: different strategies to represent the load cycles.

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Figure 12
Three-point bending test: geometry and mesh.

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Figure 13
Three-point bending test: different strategies to represent the load cycles

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Figure 14
Reinforced concrete beam - geometry and mesh.

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Figure 15
Reinforced concrete beam cyclic simulations.

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Table 1
Material parameters: direct tension.

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Table 2
Material parameters: direct compression.

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Table 3
Material parameters: three-point bending test.

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Table 4
Material parameters: concrete and steel.

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Compressive [²³23 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for plain concrete in compression," J. Am. Concr. Inst., vol. 82, no. 6, pp. 797–804, Nov. 1985, http://dx.doi.org/10.14359/10390. https://doi.org/10.14359/10390... ], [²⁴24 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for reinforced concrete in tension," J. Am. Concr. Inst., vol. 83, no. 1, pp. 21–28, Jan. 1986, http://dx.doi.org/10.14359/1756. https://doi.org/10.14359/1756... ] and tension [²⁵25 T. Boone, P. A. Wawrzynek, and A. R. Ingraffea, "Simulation of the fracture process in rock with application to hydrofracturing," Int. J. Rock Mech. Min. Sci., vol. 23, no. 3, pp. 255–265, 1986, http://dx.doi.org/10.1016/0148-9062(86)90971-X. https://doi.org/10.1016/0148-9062(86)909... ], [²⁶26 T. Boone and A. R. Ingraffea "Simulation of the fracture process at rock interfaces," in 4th Int. Conf. on Numerical Methods in Fracture Mec., 1987, pp. 519–531.] law				Linear focus law [²²22 Y. H. Lee, K. Willam, and H. D. Kang "Experimental observations of concrete behavior under uniaxial compression," in Proceedings FRAMCOS-2, 1995, pp. 397–414.], [²⁷27 Y. H. Lee and K. Willam, "Mechanical properties of concrete in uniaxial compression," ACI Mater. J., vol. 94, no. 6, pp. 457–471, Nov 1997, http://dx.doi.org/10.14359/329. https://doi.org/10.14359/329... ]
				ε_f	-0.002	σ_f	-40.4 MPa
f_c	40.4 MPa	f_t	3.53 MPa	Popovics-Saenz law [ ²⁸28 J. J. C. Popovics, "A numerical approach to the complete stress-strain curve of concrete," Cement Concr. Res., vol. 3, no. 5, pp. 583–599, Sep 1973, http://dx.doi.org/10.1016/0008-8846(73)90096-3. https://doi.org/10.1016/0008-8846(73)900... ]-[ ³¹31 B. Y. Bahn "Behavior of concrete and slender reinforced concrete columns under cyclic axial compression with bidirectional eccentricities," Ph.D. dissertation, Dept. Civ. Env. Eng., NJIT, Newwark, NJ, 1994. [Online]. Available: https://digitalcommons.njit.edu. https://digitalcommons.njit.edu... ]
ε_c	0.002	E	29,915 MPa	β_ref	1.8	k_ref	0.85
ν	0.18	G_f	0.0564 N/mm	ε_f1	-0.002	σ_f1	-40.4 MPa
h	166 mm	β_r	0.05	ε_f2	0.000118	σ_f2	3.53 MPa

Compressive [²³23 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for plain concrete in compression," J. Am. Concr. Inst., vol. 82, no. 6, pp. 797–804, Nov. 1985, http://dx.doi.org/10.14359/10390. https://doi.org/10.14359/10390... ], [²⁴24 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for reinforced concrete in tension," J. Am. Concr. Inst., vol. 83, no. 1, pp. 21–28, Jan. 1986, http://dx.doi.org/10.14359/1756. https://doi.org/10.14359/1756... ] and tension [²⁵25 T. Boone, P. A. Wawrzynek, and A. R. Ingraffea, "Simulation of the fracture process in rock with application to hydrofracturing," Int. J. Rock Mech. Min. Sci., vol. 23, no. 3, pp. 255–265, 1986, http://dx.doi.org/10.1016/0148-9062(86)90971-X. https://doi.org/10.1016/0148-9062(86)909... ], [²⁶26 T. Boone and A. R. Ingraffea "Simulation of the fracture process at rock interfaces," in 4th Int. Conf. on Numerical Methods in Fracture Mec., 1987, pp. 519–531.] law				Linear focus law [²²22 Y. H. Lee, K. Willam, and H. D. Kang "Experimental observations of concrete behavior under uniaxial compression," in Proceedings FRAMCOS-2, 1995, pp. 397–414.], [²⁷27 Y. H. Lee and K. Willam, "Mechanical properties of concrete in uniaxial compression," ACI Mater. J., vol. 94, no. 6, pp. 457–471, Nov 1997, http://dx.doi.org/10.14359/329. https://doi.org/10.14359/329... ]
				ε_f	0,000087	σ_f	2.76 MPa
f_c	27.6 MPa	f_t	2.76 MPa	Popovics-Saenz law [ ²⁸28 J. J. C. Popovics, "A numerical approach to the complete stress-strain curve of concrete," Cement Concr. Res., vol. 3, no. 5, pp. 583–599, Sep 1973, http://dx.doi.org/10.1016/0008-8846(73)90096-3. https://doi.org/10.1016/0008-8846(73)900... ]-[ ³¹31 B. Y. Bahn "Behavior of concrete and slender reinforced concrete columns under cyclic axial compression with bidirectional eccentricities," Ph.D. dissertation, Dept. Civ. Env. Eng., NJIT, Newwark, NJ, 1994. [Online]. Available: https://digitalcommons.njit.edu. https://digitalcommons.njit.edu... ]
ε_c	0.0016	E	31,700 MPa	β_ref	1.7	k_ref	0.85
ν	0.18	G_f	0.04 N/mm	ε_f1	0.000087	σ_f1	2.76 MPa
h	82.6 mm	β_r	0.05	ε_f2	-0.0016	σ_f2	-27.6 MPa

Compressive [²³23 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for plain concrete in compression," J. Am. Concr. Inst., vol. 82, no. 6, pp. 797–804, Nov. 1985, http://dx.doi.org/10.14359/10390. https://doi.org/10.14359/10390... ], [²⁴24 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for reinforced concrete in tension," J. Am. Concr. Inst., vol. 83, no. 1, pp. 21–28, Jan. 1986, http://dx.doi.org/10.14359/1756. https://doi.org/10.14359/1756... ] and tension [²⁵25 T. Boone, P. A. Wawrzynek, and A. R. Ingraffea, "Simulation of the fracture process in rock with application to hydrofracturing," Int. J. Rock Mech. Min. Sci., vol. 23, no. 3, pp. 255–265, 1986, http://dx.doi.org/10.1016/0148-9062(86)90971-X. https://doi.org/10.1016/0148-9062(86)909... ], [²⁶26 T. Boone and A. R. Ingraffea "Simulation of the fracture process at rock interfaces," in 4th Int. Conf. on Numerical Methods in Fracture Mec., 1987, pp. 519–531.] law				Popovics-Saenz law [²⁸28 J. J. C. Popovics, "A numerical approach to the complete stress-strain curve of concrete," Cement Concr. Res., vol. 3, no. 5, pp. 583–599, Sep 1973, http://dx.doi.org/10.1016/0008-8846(73)90096-3. https://doi.org/10.1016/0008-8846(73)900... ]-[³¹31 B. Y. Bahn "Behavior of concrete and slender reinforced concrete columns under cyclic axial compression with bidirectional eccentricities," Ph.D. dissertation, Dept. Civ. Env. Eng., NJIT, Newwark, NJ, 1994. [Online]. Available: https://digitalcommons.njit.edu. https://digitalcommons.njit.edu... ]
f_c	59.5 MPa	f_t	3.75 MPa	β_ref	1.8	k_ref	0.85
ε_c	0.002	E	40485 MPa	ε_f	0.0	σ_f	0.0
ν	0.2	G_f	0.115 MN/m
h	15 m	β_r	0.0001

Concrete: compressive [²³23 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for plain concrete in compression," J. Am. Concr. Inst., vol. 82, no. 6, pp. 797–804, Nov. 1985, http://dx.doi.org/10.14359/10390. https://doi.org/10.14359/10390... ], [²⁴24 D. J. Carreira and K.-H. Chu, "Stress-strain relationship for reinforced concrete in tension," J. Am. Concr. Inst., vol. 83, no. 1, pp. 21–28, Jan. 1986, http://dx.doi.org/10.14359/1756. https://doi.org/10.14359/1756... ] and tension [²⁵25 T. Boone, P. A. Wawrzynek, and A. R. Ingraffea, "Simulation of the fracture process in rock with application to hydrofracturing," Int. J. Rock Mech. Min. Sci., vol. 23, no. 3, pp. 255–265, 1986, http://dx.doi.org/10.1016/0148-9062(86)90971-X. https://doi.org/10.1016/0148-9062(86)909... ], [²⁶26 T. Boone and A. R. Ingraffea "Simulation of the fracture process at rock interfaces," in 4th Int. Conf. on Numerical Methods in Fracture Mec., 1987, pp. 519–531.] law								Steel: Von Mises criteria
f_c	20.0 MPa	f_t	2.0 MPa	ν	0.2	G_f	3.0 x 10^-5 MN/m	E	210,000 MPa
ε_c	0.002	ε_t	0.00008	β_ref	1.8	k_ref	0.85	ν	0.2
E	25,000 MPa	β_r	0.05	h	0.06 m	focus	origin	f_y	500 MPa

[{}_{l}^{s}{D]} = [\begin{matrix} {}_{l}^{s}{D_{11}} & {}_{l}^{s}{D_{12}} & {}_{l}^{s}{D_{13}} \\ {}_{l}^{s}{D_{21}} & {}_{l}^{s}{D_{22}} & {}_{l}^{s}{D_{23}} \\ {}_{l}^{s}{D_{31}} & {}_{l}^{s}{D_{32}} & {}_{l}^{s}{D_{33}} \end{matrix}] = \frac{1}{1 - \frac{E_{n} E_{t}}{E_{0}^{2}} υ^{2}} [\begin{matrix} E_{n} & \frac{υ E_{n} E_{t}}{E_{0}} & 0 \\ \frac{υ E_{n} E_{t}}{E_{0}} & E_{t} & 0 \\ 0 & 0 & (1 - \frac{E_{n} E_{t}}{E_{0}^{2}} υ^{2}) G_{n t} \end{matrix}]

\{\begin{matrix} σ_{n} \\ σ_{t} \\ τ_{n t} \end{matrix}\} = \frac{1}{1 - \frac{E_{n} E_{t}}{E_{0}^{2}} υ^{2}} [\begin{matrix} E_{n} & \frac{υ E_{n} E_{t}}{E_{0}} & 0 \\ \frac{υ E_{n} E_{t}}{E_{0}} & E_{t} & 0 \\ 0 & 0 & (1 - \frac{E_{n} E_{t}}{E_{0}^{2}} υ^{2}) G_{n t} \end{matrix}] \{\begin{matrix} ε_{n} \\ ε_{t} \\ γ_{n t} \end{matrix}\},

\frac{\partial [{}_{l}^{s}{D]}}{\partial {{ε}_{l}}} {{ε}_{l}} = [\begin{matrix} \frac{\partial {}_{l}^{s}{D_{11}}}{\partial ε_{n}} ε_{n} + \frac{\partial D_{12}}{\partial ε_{n}} ε_{t} & \frac{\partial {}_{l}^{s}{D_{11}}}{\partial ε_{t}} ε_{n} + \frac{\partial D_{12}}{\partial ε_{t}} ε_{t} & 0 \\ \frac{\partial {}_{l}^{s}{D_{21}}}{\partial ε_{n}} ε_{n} + \frac{\partial D_{22}}{\partial ε_{n}} ε_{t} & \frac{\partial {}_{l}^{s}{D_{21}}}{\partial ε_{t}} ε_{n} + \frac{\partial D_{22}}{\partial ε_{t}} ε_{t} & 0 \\ 0 & 0 & 0 \end{matrix}] .

[{}_{g}^{t}D] = \frac{\partial \{σ_{g}\}}{\partial \{ε_{g}\}} = [{}_{g}^{s}D] + {[T_{ε}]}^{T} \frac{\partial [{}_{l}^{s}D]}{\partial \{ε_{l}\}} \{ε_{l}\} [T_{ε}] + \frac{\partial {[T_{ε}]}^{T}}{\partial θ} \{σ_{l}\} {(\frac{\partial θ}{\partial \{ε_{g}\}})}^{T},

[{}_{g}^{t}D] = \frac{\partial \{σ_{g}\}}{\partial \{ε_{g}\}} = [{}_{g}^{s}D] + {[T_{ε}]}^{T} \frac{\partial [{}_{l}^{s}D]}{\partial \{ε_{l}\}} \{ε_{l}\} [T_{ε}] .

σ = f_{i} \frac{k (ε / ε_{i})}{k - 1 + {(ε / ε_{i})}^{k}}, w h e r e k = \frac{1}{1 - (f_{i} / ε_{i} . E_{0})} a n d i = t, c .

E_{G S} = \frac{1}{ε^{'}} [σ_{u} - \frac{(σ_{u} - σ_{f})}{(ε_{u} - ε_{f})} (ε_{u} - ε^{'})] .

σ = f_{i} \frac{K (ε / ε_{i})}{1 + A (ε / ε_{i}) + B {(ε / ε_{i})}^{2} + C {(ε / ε_{i})}^{3} + D {(ε / ε_{i})}^{R}}, w i t h i = c, t .

[1] Corresponding author: Lívia Ramos Santos Pereira. E-mail: lrsp@ufmg.br

Brasil

Brasil

Nonlinear analysis method of concrete structures under cyclic loading based on the generalized secant modulus

Método para análise não linear de estruturas de concreto submetidas a carregamentos cíclicos baseado no módulo secante generalizado

Abstract