Comparative analysis of multilinear constitutive models of steel fiber reinforced concrete

Fasolo, Flávia; Kroetz, Henrique Machado; Pieralisi, Ricardo

doi:10.1590/S1983-41952024000300009

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ORIGINAL ARTICLE • Rev. IBRACON Estrut. Mater. 17 (3) • 2024 • https://doi.org/10.1590/S1983-41952024000300009 copy

This document is related to:

10.1590/S1983-41952024000300011

Comparative analysis of multilinear constitutive models of steel fiber reinforced concrete

Análise comparativa de modelos constitutivos multilineares de concreto reforçado com fibras de aço

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The challenging task of structural design under structural safety, economic efficiency and environmental sustainability constraints has led to the development of several efficient materials. In this context, steel fiber reinforced concrete (SFRC) stands out due to the post-cracking behavior related to its enhanced toughness. Recent structural design normative codes consider this composite, suggesting good structural performance and broadening its applicability. Nevertheless, literature still lacks a comprehensive constitutive model to precisely describe its tensile behavior. This paper analyzes and compares three of the main European constitutive models, investigating factors that possibly influence their behavior. This study was conducted based on the cross-sectional analysis method, so that load-CMOD graphs could be obtained. Statistical analysis was conducted to classify the models, regarding several relevant parameters, such as mean compressive strength of the matrix and fiber volume. Results show that the constitutive law proposed by the fib Model Code shows the best performance. Also, from the set of considered parameters, ultimate tensile fibers strength presents the greatest sensitivity concerning structural post-crack behavior.

Keywords:
fiber reinforced concrete; constitutive equations; sectional analysis; steel fiber

Diagram	Parameters	Reference
	$σ_{1} = {0.7 ∙ f c t m}_{f l} ∙ (1.6 - d)$	RILEM [¹⁷17 RILEM, "RILEM TC 162-TDF: test and design methods for steel fibre reinforced concrete sigma-epsilon-design method - final recommendation," Mater. Struct., vol. 36, pp. 560-567, 2003.]
	$σ_{2} = {0.45 ∙ K}_{h} ∙ f_{R, 1}$
	$σ_{3} = {0.37 ∙ K}_{h} ∙ f_{R, 4}$
	$ε_{1} = {σ_{1} / E}_{H R F}$
	$ε_{2} = ε_{2} + 0.1 ‰$
	$ε_{3} = ε_{u} = 25 ‰$
	$f_{F t s} = 0.45 \times f_{R 1}$	MC2010 [¹⁸18 International Federation for Structural Concrete, fib Bulletin 65: Model Code for Concrete Structures, Final Draft: 5.6 Fibres/Fibre Reinforced Concrete, vol. 1, 2010.]
	$f_{F t u} = k \times [f_{F t s} - (w_{u} / {C M O D}_{3}) \times (f_{F t s} - 0.5 \times f_{R 3} + 0.2 \times f_{R 1})]$
	$ε_{S L S} = {C M O D}_{1} / l_{c s}$
	$ε_{S L U} = \frac{w_{u}}{l_{c s}} = \min (ε_{F u} = \frac{2.5}{l_{c s}} = \frac{2.5}{y})$
	$ε_{F u} = (20 ‰ s o f t e n i n g; 10 ‰ h a r d e n i n g)$
	$σ_{1} = f_{c t, d} = 0.6 \times f_{c t, f l, d}$	CE2021 [⁶6 España, “Código Estructural: Anejo 7: Recomendaciones para la utilización de hormigón con fibras, CE," Bol. Of. Estado, 2021.]
	$σ_{2} = f_{c t R 1, d} = 0.45 \times f_{R, 1, d}$
	$σ_{3} = f_{c t R 3, d} = k_{1} \times (0.5 \times f_{R, 3, d} - 0.2 \times f_{R, 1, d})$
	$ε_{2} = 0.1 + 1000 \times f_{c t, d} / E_{c, 0}$
	$ε_{3} = 2.5 / l_{c s} (l c s : c h a r a c t e r i s t i c l e n g h t)$
	$ε_{u} = (20 ‰ b e n d i n g; 10 ‰ p u r e t e n s i o n)$

Parameter	Group 1	Group 2
Mean compressive strength, MPa	≤50	> 50
Volume fraction of fibers, %	≤ 0.5	> 0.5
Fiber aspect ratio	≤ 60	> 60
Fiber tensile strength, MPa	≤ 1345	> 1345

Identification		Flop	F1	F2	F3	F4
≤50 MPa	RILEM	0.40 (0.14)	0.20 (0.05)	0.16 (0.08)	0.18 (0.1)	0.24 (0.1)
	MC2010	-0.01 (0.14)	0.17 (0.04)	0.13 (0.15)	0.15 (0.25)	0.21 (0.29)
	CE2021	0.09 (0.14)	0.25 (0.04)	0.15 (0.07)	0.07 (0.04)	0.18 (0.08)
>50 MPa	RILEM	0.56 (0.15)	0.19 (0.04)	0.1 (0.06)	0.1 (0.06)	0.18 (0.07)
	MC2010	0.04 (0.14)	0.15 (0.03)	0.02 (0.10)	-0.03 (0.16)	0.02 (0.20)
	CE2021	0.08 (0.16)	0.22 (0.04)	0.09 (0.06)	0.03 (0.03)	0.13 (0.06)
≤0.5%	RILEM	0.54 (0.16)	0.23 (0.04)	0.12 (0.08)	0.12 (0.08)	0.20 (0.12)
	CE2021	-0.02 (0.13)	0.18 (0.03)	0.05 (0.14)	-0.01 (0.19)	0.04 (0.24)
	MC2010	0.03 (0.15)	0.26 (0.04)	0.11 (0.08)	0.04 (0.04)	0.14 (0.08)
>0.5%	RILEM	0.47 (0.15)	0.17 (0.03)	0.11 (0.05)	0.13 (0.07)	0.20 (0.08)
	CE2021	0.05 (0.11)	0.13 (0.02)	0.05 (0.10)	0.04 (0.18)	0.10 (0.24)
	MC2010	0.12 (0.15)	0.21 (0.03)	0.10 (0.05)	0.04 (0.03)	0.15 (0.07)
≤60	RILEM	0.47 (0.14)	0.20 (0.04)	0.13 (0.06)	0.14 (0.08)	0.23 (0.09)
	CE2021	-0.04 (0.09)	0.16 (0.03)	0.09 (0.13)	0.06 (0.18)	0.16 (0.27)
	MC2010	0.02 (0.13)	0.25 (0.04)	0.13 (0.06)	0.06 (0.04)	0.17 (0.08)
>60	RILEM	0.55 (0.19)	0.19 (0.05)	0.1 (0.07)	0.11 (0.08)	0.17 (0.08)
	CE2021	0.06 (0.13)	0.15 (0.04)	0.02 (0.12)	-0.02 (0.19)	0.03 (0.23)
	MC2010	0.10 (0.14)	0.22 (0.04)	0.09 (0.07)	0.03 (0.05)	0.12 (0.07)
≤1345 MPa	RILEM	0.49 (0.18)	0.21 (0.04)	0.16 (0.08)	0.16 (0.09)	0.23 (0.09)
	CE2021	-0.04 (0.11)	0.17 (0.03)	0.11 (0.14)	0.10 (0.23)	0.15 (0.26)
	MC2010	0.02 (0.11)	0.26 (0.04)	0.15 (0.07)	0.06 (0.04)	0.17 (0.08)
>1345 MPa	RILEM	0.55 (0.16)	0.19 (0.03)	0.08 (0.05)	0.08 (0.05)	0.15 (0.06)
	CE2021	0.04 (0.13)	0.14 (0.03)	-0.02 (0.09)	-0.1 (0.14)	-0.06 (0.18)
	MC2010	0.08 (0.17)	0.21 (0.03)	0.07 (0.05)	0.02 (0.03)	0.11 (0.07)

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E-mail: arlene@ibracon.org.br

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[1] Corresponding author: Flávia Fasolo. E-mail: flaviarfasolo@gmail.com