Leis constitutivas cíclicas baseadas na proposta de ponto focal móvel para concreto simples submetido à tensão e compressão

Pereira, Lívia Ramos Santos; Penna, Samuel Silva

Lívia Ramos Santos Pereira

conceptualization

formal analysis

methodology

writing

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

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

Samuel Silva Penna

conceptualization

methodology

writing

supervision

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

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

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

Conflict of interest: Nothing to declare.

Editors: Samir Maghous, Daniel Carlos Taissum Cardoso.

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Figure 1
Stress-strain laws for concrete under compression (a) and tension (b).

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Figure 2
Principal strain directions and the global system.

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Figure 3
Cyclic approaches based on a fixed focal point.

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Figure 4
Model based on multiple focal points (adapted from Yankelevsky and Reinhardt [⁵5 D. Z. Yankelevsky and H. W. Reinhardt, “Focal point model for uniaxial cyclic behavior of concrete,” in Proc. IABSE Colloq., Delft, 1987, pp. 99–106, http://doi.org/10.5169/seals-41920.
http://doi.org/10.5169/seals-41920... ]).

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Figure 5
Stress-strain law with moving focal point and linear unloading-reloading.

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Figure 6
Nonlinear cyclic approach.

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Figure 7
R multiplier: parameter analysis.

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Figure 8
Structural models.

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Figure 9
Tension test with the linear cyclic approach.

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Figure 10
Tension test with nonlinear cyclic approach.

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Figure 11
Compression test with linear cyclic approach.

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Figure 12
Compression test with nonlinear cyclic approach.

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Figure 13
Damage evolution and secant modulus variation.

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Figure 9
Tension test with the linear cyclic approach.

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Figure 10
Tension test with nonlinear cyclic approach.

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Figure 11
Compression test with linear cyclic approach.

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Figure 12
Compression test with nonlinear cyclic approach.

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Figure 13
Damage evolution and secant modulus variation.

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Table 1
Focal points stress coordinates.

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Table 2
Relation among the auxiliaries points and the stress intervals.

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Table 3
Study of the variable

Δ κ

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Point	Coordinate	Point	Coordinate	Point	Coordinate
$C_{1}$	$3 f_{c}$	$C_{2}$	$f_{c}$	$C_{3}$	$0.75 f_{c}$
$C_{4}$	$0.47 f_{c}$	$C_{5}$	$0.2 f_{c}$	$C_{6}$	$0.0; ε = ε_{0}$
$T_{0}$	$0.0$	$T_{1}$	$- 3 f_{t}$	$T_{2}$	$- f_{t}$
$T_{3}$	$- 0.75 f_{t}$	$T_{4}$	$- 0.5 f_{t}$	$T_{5}$	$- 0.125 f_{t}$

Condition	$σ_{a u x}$	$ε_{a u x}$	Auxiliary point
$σ_{u} > 0.75 f_{t}$	$- f_{t}$	$- f_{t} / E^{0}$	$T_{2}$
$0.75 f_{t} > σ_{u} > 0.25 f_{t}$	$- 0.5 f_{t}$	$- {0.5 f}_{t} / E^{0}$	$T_{4}$
$σ_{u} < 0.75 f_{t}$	$- 0.125 f_{t}$	$- {0.125 f}_{t} / E^{0}$	$T_{5}$
$σ_{u} > 0.75 f_{c}$	$- f_{c}$	$- f_{c} / E^{0}$	$C_{2}$
$0.75 f_{c} > σ_{u} > 0.375 f_{c}$	$- 0.47 f_{c}$	$- 0.47 f_{c} / E^{0}$	$C_{4}$
$σ_{u} < 0.375 f_{c}$	$- 0.2 f_{c}$	$- 0.2 f_{c} / E^{0}$	$C_{5}$

Tensile					Compression
Reference	Cycle	$ε_{u}^{1} (%)$	$ε_{r}^{1} (%)$	$Δ κ = ε_{r} / ε_{u}$	Reference	Cycle	$ε_{u}^{1} (%)$	$ε_{r}^{1} (%)$	$Δ κ = ε_{r} / ε_{u}$
[³²32 Y. Tanigawa and Y. Uchida, “Hysteretic characteristics of concrete in the domain of high compressive strain,” in Proc. Ann. Conv. AIJ, 1979, pp. 449–450.]	1	0.128	0.134	1.047	[²2 I. D. Karsan and S. O. Jirsa, "Behavior of concrete under compressive loadings," J. Struct. Div., vol. 95, no. 12, pp. 2543–2563, 1969, http://doi.org/10.1061/JSDEAG.0002424. http://doi.org/10.1061/JSDEAG.0002424... ] 1^st test	1	0.865	1.137	1.3142
	2	0.186	0.201	1.080		2	1.137	1.407	1.238
	3	0.262	0.278	1.061		3	1.407	1.587	1.128
	4	0.321	0.340	1.059		4	1.966	2.336	1.188
	5	0.393	0.415	1.057		5	2.655	2.942	1.109
[³3 H. W. Reinhardt, "Fracture mechanics of an elastic softening material like concrete," Heron, vol. 29, no. 2, pp. 1–42, 1984.], [³³33 V. Gopalaratnam and S. Shah, "Softening response of plain concrete in direct tension," J. Am. Concr. Inst., vol. 82, no. 3, pp. 310–323, 1985, http://doi.org/10.14359/10338. http://doi.org/10.14359/10338... ], [³⁴34 J. M. Moelands, “Modelling of fatigue behavior of plain concrete under alternating and repeated loading,” Ph.D. dissertation, Delft Univ., Delft, Netherlands, 1984.] 1^st test	1	0.011	0.012	1.122	[²2 I. D. Karsan and S. O. Jirsa, "Behavior of concrete under compressive loadings," J. Struct. Div., vol. 95, no. 12, pp. 2543–2563, 1969, http://doi.org/10.1061/JSDEAG.0002424. http://doi.org/10.1061/JSDEAG.0002424... ] 2^nd test	1	0.808	0.981	1.214
	2	0.016	0.020	1.243		2	1.115	1.253	1.124
	3	0.03	0.035	1.099		3	1.378	1.603	1.163
	4	0.044	0.046	1.042		4	1.603	1.854	1.156
	5	0.064	0.070	1.088		5	1.934	2.238	1.158
	6	0.106	0.110	1.034	[³⁰30 D. C. Spooner and J. W. Dougill, "A qualitative assessment of damage sustained in concrete during compressive loading," Mag. Concr. Res., vol. 27, no. 9, pp. 151–160, 1975, http://doi.org/10.1680/macr.1975.27.92.151. http://doi.org/10.1680/macr.1975.27.92.1... ]	1	1.914	2.168	1.133
[³3 H. W. Reinhardt, "Fracture mechanics of an elastic softening material like concrete," Heron, vol. 29, no. 2, pp. 1–42, 1984.], [³³33 V. Gopalaratnam and S. Shah, "Softening response of plain concrete in direct tension," J. Am. Concr. Inst., vol. 82, no. 3, pp. 310–323, 1985, http://doi.org/10.14359/10338. http://doi.org/10.14359/10338... ], [³⁴34 J. M. Moelands, “Modelling of fatigue behavior of plain concrete under alternating and repeated loading,” Ph.D. dissertation, Delft Univ., Delft, Netherlands, 1984.] 2^nd test	1	0.007	0.008	1.090		2	2.357	2.756	1.169
	2	0.008	0.008	1.016		3	4.249	4.373	1.030
	3	0.009	0.010	1.074	[³¹31 S. Okamoto, S. Shiomi, and K. Yamabe, “Earthquake resistance of pre-stressed concrete structures,” in Proc. Ann. Conv. AIJ, 1976, pp. 1251–1252.] 1^st test	1	1.965	2.141	1.090
	4	0.010	0.012	1.207		2	2.859	3.239	1.0133
	5	0.014	0.014	1.036		3	3.930	3.239	1.059
	6	0.016	0.017	1.083		4	4.838	4.162	1.086
	7	0.0020	0.023	1.145	[³¹31 S. Okamoto, S. Shiomi, and K. Yamabe, “Earthquake resistance of pre-stressed concrete structures,” in Proc. Ann. Conv. AIJ, 1976, pp. 1251–1252.] 2^nd test	1	1.801	1.976	1.097
	8	0.028	0.031	1.086		2	2.518	2.476	1.090
[³3 H. W. Reinhardt, "Fracture mechanics of an elastic softening material like concrete," Heron, vol. 29, no. 2, pp. 1–42, 1984.], [³³33 V. Gopalaratnam and S. Shah, "Softening response of plain concrete in direct tension," J. Am. Concr. Inst., vol. 82, no. 3, pp. 310–323, 1985, http://doi.org/10.14359/10338. http://doi.org/10.14359/10338... ], [³⁴34 J. M. Moelands, “Modelling of fatigue behavior of plain concrete under alternating and repeated loading,” Ph.D. dissertation, Delft Univ., Delft, Netherlands, 1984.] 3^rd test	1	0.009	0.001	1.212		3	3.550	3.953	1.113
	2	0.014	0.016	1.125		4	4.399	4.962	1.105
	3	0.019	0.021	1.148		5	5.308	5.920	1.115
	4	0.021	0.023	1.077		6	6.243	6.497	1.041
	5	0.026	0.030	1.126	[³²32 Y. Tanigawa and Y. Uchida, “Hysteretic characteristics of concrete in the domain of high compressive strain,” in Proc. Ann. Conv. AIJ, 1979, pp. 449–450.]	1	1.046	1.219	1.165
	6	0.035	0.036	1.044		2	1.614	1.660	1.029
	7	0.048	0.051	1.063		3	2.096	2.388	1.140
	8	0.075	0.077	1.028		4	2.676	2.858	1.068
						5	3.223	3.281	1.018
						6	3.651	3.997	1.095
					[¹1 B. P. Sinha, K. H. Gerstle, and L. G. Tulin, "Stress-strain relations for concrete under cyclic loading," J. Am. Concr. Inst., vol. 61, no. 2, pp. 195–211, 1964, http://doi.org/10.14359/7775. http://doi.org/10.14359/7775... ]	1	0.884	1.147	1.298
						2	1.282	1.501	1.171
						3	1.656	1.878	1.134
						4	1.878	2.052	1.093
						5	2.265	2.477	1.094
						6	2.581	2.753	1.067

[D_{g}^{t}] = {{[T}_{ε}]}^{T} ({[D}_{l}^{s}] + \frac{\partial {[D}_{l}^{s}]}{\partial {ε_{l}}} {{ε}_{l}}) {[T}_{ε}] + \frac{\partial {{[T}_{ε}]}^{T}}{\partial θ} {{σ}_{l}} \frac{\partial θ}{\partial {ε_{g}}} .

{[D}_{l}^{s}] = \frac{1}{1 - \frac{ψ}{E_{0}} ν} [\begin{matrix} E_{1} (ε_{1}) & ψ & 0 \\ ψ & E_{2} (ε_{2}) & 0 \\ 0 & 0 & (1 - \frac{ψ}{E_{0}} ν) G_{12} (ε_{1}, ε_{2}) \end{matrix}],

G_{12} (ε_{1}, ε_{2}) = \frac{E_{0} E_{1} (ε_{1}) E_{2} (ε_{2})}{E_{0} E_{1} (ε_{1}) + E_{0} E_{2} (ε_{2}) + 2 ν E_{1} (ε_{1}) E_{2} (ε_{2})},

\frac{\partial [D_{l}^{s}]}{\partial {{ε}_{l}}} {{ε}_{l}} = [\begin{matrix} \frac{\partial_{l}^{s} D_{11}}{\partial ε_{1}} ε_{1} + \frac{\partial_{l}^{s} D_{12}}{\partial ε_{1}} ε_{2} & \frac{\partial_{l}^{s} D_{11}}{\partial ε_{2}} ε_{1} + \frac{\partial_{l}^{s} D_{12}}{\partial ε_{2}} ε_{2} & 0 \\ \frac{\partial_{l}^{s} D_{12}}{\partial ε_{1}} ε_{1} + \frac{\partial_{l}^{s} D_{22}}{\partial ε_{1}} ε_{2} & \frac{\partial_{l}^{s} D_{12}}{\partial ε_{2}} ε_{1} + \frac{\partial_{l}^{s} D_{22}}{\partial ε_{2}} ε_{2} & 0 \\ 0 & 0 & 0 \end{matrix}],

{[T}_{ε}] = [\begin{matrix} \cos^{2} θ & \sin^{2} θ & \cos θ \sin θ \\ \sin^{2} θ & \cos^{2} θ & - \cos θ \sin θ \\ - 2 \cos θ \sin θ & 2 \cos θ \sin θ & \cos^{2} θ - \sin^{2} θ \end{matrix}],

\frac{\partial {{[T}_{l}]}^{T}}{\partial θ} = [\begin{matrix} - 2 \cos θ \sin θ & 2 \cos θ \sin θ & - 2 (\cos^{2} θ - \sin^{2} θ) \\ 2 \cos θ \sin θ & - 2 \cos θ \sin θ & 2 (\cos^{2} θ - \sin^{2} θ) \\ \cos^{2} θ - \sin^{2} θ & - (\cos^{2} θ - \sin^{2} θ) & - 4 \cos θ \sin θ \end{matrix}] .

\frac{\partial θ}{\partial {{ε}_{g}}} = \frac{1}{2} \frac{γ_{x y}}{{(ε_{x} - ε_{y})}^{2} + γ_{x y}^{2}} [\begin{matrix} - 1 \\ 1 \\ \frac{ε_{x} - ε_{y}}{γ_{x y}} \end{matrix}] .

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

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

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Leis constitutivas cíclicas baseadas na proposta de ponto focal móvel para concreto simples submetido à tensão e compressão

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