Cyclic constitutive laws based on the moving focal point proposal for plain concrete under tension and compression

Pereira, Lívia Ramos Santos; Penna, Samuel Silva

doi:10.1590/S1983-41952024000200012

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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Cyclic constitutive laws based on the moving focal point proposal for plain concrete under tension and compression

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

Abstract