Predição da capacidade última de elementos em concreto armado usando metodologias de análises não linear

Palomo, Ingrid Rocio Irreño; Martínez, Juan de Jesus; Benedetty, Carlos Alberto; Almeida, Luiz Carlos de; Trautwein, Leandro Mouta; Krahl, Pablo Augusto

Ingrid Rocio Irreño Palomo

conceptualization

methodology

investigation

writing original draft

Universidade Estadual de Campinas – UNICAMP, Faculdade de Engenharia Civil, Departamento de Estruturas, Campinas, SP, Brasil

https://orcid.org/0000-0002-4517-0076

Juan de Jesus Martínez

conceptualization

methodology

writing original draft

Universidade Estadual de Campinas – UNICAMP, Faculdade de Engenharia Civil, Departamento de Estruturas, Campinas, SP, Brasil

https://orcid.org/0000-0002-5181-9572

Carlos Alberto Benedetty

conceptualization

writing original draft

Universidade Estadual de Campinas – UNICAMP, Faculdade de Engenharia Civil, Departamento de Estruturas, Campinas, SP, Brasil

https://orcid.org/0000-0001-7885-0572

Luiz Carlos de Almeida

writing review and editing

supervision

project administration

funding acquisition

Universidade Estadual de Campinas – UNICAMP, Faculdade de Engenharia Civil, Departamento de Estruturas, Campinas, SP, Brasil

https://orcid.org/0000-0001-8431-8929

Leandro Mouta Trautwein

writing review and editing

supervision

project administration

funding acquisition

Universidade Estadual de Campinas – UNICAMP, Faculdade de Engenharia Civil, Departamento de Estruturas, Campinas, SP, Brasil

https://orcid.org/0000-0002-4631-9290

Pablo Augusto Krahl

writing review and editing

Universidade Presbiteriana Mackenzie – UPM, Faculdade de Engenharia Civil, Departamento de Estruturas, Campinas, SP, Brasil

https://orcid.org/0000-0002-6172-5481

Corresponding author: Ingrid Rocio Irreño Palomo. E-mail: ingridpal2393@gmail.com

Conflict of interest: Nothing to declare.

Author contributions: IRIP: conceptualization, methodology, investigation, writing - original draft. JJM: conceptualization, methodology, writing - original draft. CAB: Conceptualization, Writing - original draft. LCA: writing - review & editing, supervision, project administration, funding acquisition. LMT: writing - review & editing, supervision, project administration, funding acquisition. PAK: writing - review & editing.

Editors: Osvaldo Manzoli, Guilherme Aris Parsekian.

Parameter	Concrete	Long. Steel	Transv. Steel	Reference
Elastic Modulus - $E$	23674 MPa	192500 MPa	200000 MPa	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Hardening modulus - $E_{s h}$	/	3100 MPa	3100 MPa	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Fracture energy - $G_{f}$	90 N/m	/	/	[³⁵35 Comité Euro-International du Béton, Fédération Internationale du Béton, fib Model Code for Concrete Structures 2010, CEB-FIB 2010, 2010.]
Shear factor - $S_{f}$	50	/	/	adopted
Compression strength - $f_{c}$	30 MPa	/	/	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Tensile strength - $f_{t}$	1.81 MPa	/	/	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Strength at onset of nonlinear behavior - $f_{c o}$	3.6 MPa	/	/	[³3 V. Červenka, L. Jendele, and J. Červenka, ATENA Program Documentation Part 1: Theory. Prague: Červenka Consulting, 2021.]
Plastic strain at compressive strength - ${ε_{c}^{p}}_{}^{}$	0.00185	/	/	[³⁴34 Comité Euro-International du Béton, Fédération Internationale du Béton, fib Model Code for Concrete Structures 1990, CEB-FIB 90, 1990, https://doi.org/10.1680/ceb-fipmc1990.35430. https://doi.org/10.1680/ceb-fipmc1990.35... ]
Reduction factor of the compressive strength	0.8	/	/	[³3 V. Červenka, L. Jendele, and J. Červenka, ATENA Program Documentation Part 1: Theory. Prague: Červenka Consulting, 2021.]
Yield strength - $f_{y}$	/	418 MPa	596 MPa	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Ultimate strength - $f_{u}$	/	454 MPa	640 MPa	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Yield strain - $ε_{s}$	/	0.00217	0.00298	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Strain hardening - $ε_{s h}$	/	0.0095	0.0095	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]
Ultimate strain - $ε_{u}$	/	0.0669	0.0695	[²⁷27 F. J. Vecchio and M. B. Emara, “Shear deformations in reinforced concrete frames,” ACI Struct. J., vol. 89, no. 1, pp. 46–56, Jan. 1992, http://dx.doi.org/10.14359/1283. http://dx.doi.org/10.14359/1283... ]

Model	Mesh size	Ultimate load [kN]	Error [%]
Experimental test		332	---
Coarse mesh	100 mm	345.1	3.8
Medium mesh	80 mm	340.9	2.6
Fine mesh	50 mm	332.8	0.24

Element	Numerical result -NLFEA	Numerical result -NLSA	Experimental result
Flexural cracks in the beam B1	59.4 kN (Figure 17a)	61.6 kN – elements 1 and 14, stiffness: 41% (Figure 17b)	52.5 kN
Flexural cracks at the base of the columns	152.8 kN (Figure 18a)	145 kN – element 1, column C1, stiffness: 43%	145 kN
Flexural cracks at the base of the columns	152.8 kN (Figure 18a)	145 kN – element 1, column C2, stiffness: 50% (Figure 18b)	145 kN

Element	Numerical result -NLFEA	Numerical result -NLSA	Experimental result
Yielding in the longitudinal reinforcement of beam B1	281.5 kN (Figure 19a)	302.4 kN – element 1, stiffness: 0.75%	264 kN (top)
Yielding in the longitudinal reinforcement of beam B1	281.5 kN (Figure 19a)	302.4 kN – element 14, stiffness: 0.85% (Figure 19b)	287 kN (bottom)
Yielding in the longitudinal reinforcement of column	326.1 kN (Figure 20a)	320 kN – element 1, column C1, stiffness: 0.81%	323 kN
Yielding in the longitudinal reinforcement of column	326.1 kN (Figure 20a)	335.2 kN – element 1, column C2, stiffness: 2.7% (Figure 20b)	323 kN
Yielding in the longitudinal reinforcement of beam B2	326.1 kN (Figure 20a)	340.8 kN – element 1, stiffness: 1.0%	329 kN
Yielding in the longitudinal reinforcement of beam B2	326.1 kN (Figure 20a)	340.8 kN – element 14, stiffness: 0.75% (Figure 21)	329 kN

\frac{σ}{f_{t}} = \{1 + {(c_{1} \frac{w}{w_{c}})}^{3}\} e x p (- c_{2} \frac{w}{w_{c}}) - \frac{w}{w_{c}} (1 + {c_{1}}^{3}) e x p (- c_{2})

K = [\begin{matrix} \begin{matrix} \frac{1}{C_{1}} & 0 & 0 \\ 0 & \frac{C_{3}}{C} & \frac{C_{2}}{C} \\ 0 & \frac{C_{2}}{C} & \frac{C_{2} L - C_{5}}{C} \end{matrix} & \begin{matrix} \frac{- 1}{C_{1}} & 0 & 0 \\ 0 & \frac{- C_{3}}{C} & \frac{C_{6}}{C} \\ 0 & \frac{- C_{2}}{C} & \frac{C_{5}}{C} \end{matrix} \\ \begin{matrix} \frac{- 1}{C_{1}} & 0 & 0 \\ 0 & \frac{- C_{3}}{C} & \frac{- C_{2}}{C} \\ 0 & \frac{C_{6}}{C} & \frac{C_{5}}{C} \end{matrix} & \begin{matrix} \frac{1}{C_{1}} & 0 & 0 \\ 0 & \frac{C_{3}}{C} & \frac{- C_{6}}{C} \\ 0 & \frac{- C_{6}}{C} & \frac{C_{6} L - C_{5}}{C} \end{matrix} \end{matrix}]

C_{5} = \frac{l^{3}}{6} \sum_{i = 1}^{n} \frac{1}{E_{i} I_{i}} + {\frac{l^{3}}{2} \sum_{i = 1}^{n} \frac{n - 1}{E_{i} I_{i}} + l}^{3} \sum_{i = 1}^{n - 2} \frac{(n - 1) i - i^{2}}{E_{i + 1} I_{i + 1}}

M = k_{1} f_{c}^{'} b k d (\frac{h}{2} - k_{2} k d) + \sum_{i = 1}^{2} σ_{s i} A_{s i} (\frac{h}{2} - d_{i}) + k_{3} f_{t}^{'} b (h - k d) [\frac{h}{2} - k_{4} (h - k d)]

[1] Corresponding author: Ingrid Rocio Irreño Palomo. E-mail: ingridpal2393@gmail.com

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Predição da capacidade última de elementos em concreto armado usando metodologias de análises não linear

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