Prediction of the ultimate capacity of reinforced concrete elements using nonlinear analysis methodologies

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

doi:10.1590/S1983-41952024000200010

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

Brasil

Brasil

Prediction of the ultimate capacity of reinforced concrete elements using nonlinear analysis methodologies

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

Abstract