Aspects of Finite Element Modeling of Punching Shear Behavior of Reinforced Concrete Flat Slabs

D., Rafael A. Sanabria; Oliveira, Leonardo H. B. de; Trautwein, Leandro Mouta; Almeida, Luiz Carlos de; Santos, Antônio Carlos dos

doi:10.1590/1679-78255004

Rafael A. Sanabria D. ^**Corresponding authors

Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas - UNICAMP, Campinas, São Paulo, Brazil. E-mail: rafael_sanabria25@hotmail.com, leandromt@fec.unicamp.br, almeida@fec.unicamp.br, leonardo.oliveira@ufersa.edu.br

Leonardo H. B. de Oliveira ^**Corresponding authors

Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas - UNICAMP, Campinas, São Paulo, Brazil. E-mail: rafael_sanabria25@hotmail.com, leandromt@fec.unicamp.br, almeida@fec.unicamp.br, leonardo.oliveira@ufersa.edu.br

Universidade Federal Rural do Semi-Árido - UFERSA, Pau dos Ferros, Rio Grande do Norte, Brazil.

Leandro Mouta Trautwein

Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas - UNICAMP, Campinas, São Paulo, Brazil. E-mail: rafael_sanabria25@hotmail.com, leandromt@fec.unicamp.br, almeida@fec.unicamp.br, leonardo.oliveira@ufersa.edu.br

Luiz Carlos de Almeida

Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas - UNICAMP, Campinas, São Paulo, Brazil. E-mail: rafael_sanabria25@hotmail.com, leandromt@fec.unicamp.br, almeida@fec.unicamp.br, leonardo.oliveira@ufersa.edu.br

Antônio Carlos dos Santos

Faculdade de Engenharia Civil, Universidade Federal de Uberlândia - UFU, Uberlândia, Minas Gerais, Brazil. E-mail: acds@feciv.ufu.br

*Corresponding authors

	Concrete Properties
Parameter	Baseline Analysis	Observations
Young’s Modulus ( $E$ )	28 800 MPa	Calculated according to CEB-FIP MC1990 (Comité Euro-International du Béton, 1993Comité Euro-International du Béton. CEB-FIP Model Code 1990-Final draft, Lausanne, Thomas Telford, 1993.)
Poisson Ration ( $ν$ )	0.2	-
Crack Model	Total Strain Fixed Model	-
Tensile Strength ( $f_{t}$ )	3.0 MPa	Calculated according to CEB-FIP MC1990 (Comité Euro-International du Béton, 1993Comité Euro-International du Béton. CEB-FIP Model Code 1990-Final draft, Lausanne, Thomas Telford, 1993.)
Tensile Softening Curve	Hordijk (1991Hordijk, D. (1991). Local Approach to Fracture of Concrete. Ph. D Thesis. Delft University of Technology, Delft, The Netherlands.)	-
Fracture Tensile Energy ( $G_{f}$ )	0.141 N/mm	Calculated according to fib MC2010 (Comité Euro-International du Béton, 2012Comité Euro-International du Béton. fib Model Code 2010-Final draft. Lausanne, 2012.)
Compressive Curve	Parabolic	Recommended by Hendriks et al. (2016Hendriks, M.; Boer, A. de; Belletti, B. (2016). Guidelines for nonlinear finite element analysis of concrete structures. Rijkswaterstaat Technisch Document (RTD), Rijkswaterstaat Centre for Infrastructure, RTD, v. 1016, p. 2012.)
Shear Behavior	Damage Based Shear Retention	The Shear stiffness decays of the shear stress beyond the cut-off stress
Compressive Strength	39.1 MPa	Ramos (2003Ramos, A. M. P. (2003.) Punching in prestressed concrete flat slab. Ph. D Thesis (in Portuguese). Universidade Técnica de Lisboa-Instituto Superior Técnico, Lisboa, Portugal.) Experimental test results
Reduction of Compressive strength due to lateral crack	Vecchio and Collins (1993Vecchio, F. J.; Collins, M. P. (1993). Compression response of cracked reinforced concrete. Journal of Structural Engineering, American Society of Civil Engineers, v. 119, n. 12, p. 3590-3610.)	-
Confinement Model	Selby and Vecchio (1997Selby, R.; Vecchio, F. (1997). A constitutive model for analysis of reinforced concrete solids. Canadian Journal of Civil Engineering, NRC Research Press, v. 24, n. 3, p. 460-470.)	-
Reduction of Poisson Ratio	Included	-
Compressive Fracture Energy ( $G_{c}$ )	35.31 N/mm	$G_{c} = 250 G_{f}$ , recommended by *Nakamura and Higai (2001* Nakamura, Hikaru; Higai, Takeshi. (2001). Compressive fracture energy and fracture zone length of concrete. Modeling of inelastic behavior of RC structures under seismic loads, 471-487. )

	Convergence Parameters
Parameter	Baseline Analysis	Observations
Iteration Solution Procedure	Quasi-Newton (Secant)-BFGS	-
Maximum Number of Iteration per step	200	-
Incremental-iterative Procedure	Displacement Control	-
Loading incremental step size	0.2 mm	-
Convergence Criteria	Force Norm (0.01) Displacement Norm (0.01) Energy Norm (0.001)	Default Values of the Software

Line Search Technique	Included	Recommended for improving the convergence rate of iteratives procedure

Mesh Type	Layers on the slab Thickness	Elapsed time	Peak Load (kN)	Displacement at Peak Load (mm)
D	4	01:32:58	243.45	19.8
G	4	01:05:30	243.62	18.4
E	5	02:20:00	261.29	22.2
H	5	01:42:36	261.64	22.0

Model	Convergence Method	Criteria Tolerance			Load incremental step size (mm/step)
Model	Convergence Method	Force	Displacement	Energy
CM01	Regular Newton Raphson	$10^{- 2}$	$10^{- 2}$	$10^{- 4}$	0.2
CM02	Modified Newton-Raphson	$10^{- 2}$	$10^{- 2}$	$10^{- 4}$	0.2
CM03	Secant	N/A	N/A	$10^{- 4}$	0.2
CM04	Secant	$10^{- 2}$	N/A	N/A	0.2
CM05	Secant	N/A	$10^{- 2}$	N/A	0.2
CM06	Secant	$10^{- 2}$	$10^{- 2}$	$10^{- 4}$	0.2
CM07	Secant	N/A	N/A	$10^{- 4}$	0.5
CM08	Secant	N/A	N/A	$10^{- 4}$	1.0
CM09	Secant	N/A	N/A	$10^{- 3}$	0.2
CM10	Secant	N/A	N/A	$10^{- 2}$	0.2
CM11	Secant	N/A	N/A	$10^{- 4}$	0.2

Model	Elapsed time	Peak Load (kN)	Displacement at Peak Load (mm)	Number of steps Converged	Observations
CM01	01:30:54	244.82	18.2	95	Regular Newton-Raphson
CM02	00:37:57	234.39	16.8	85	Modified Newton-Raphson
CM03	00:54:26	258.87	21.2	109	Secant (E)
CM04	00:40:24	259.06	21.2	108	Secant (F)
CM05	02:52:43	228.28	16.0	115+	Secant (D)
CM06	01:42:36	261.64	22.0	111	Baseline: Secant (F D E)
CM07	00:29:04	262.74	22.5	46	Step size: 0.5 mm/step
CM08	00:22:06	265.53	23.0	25	Step Size: 1.0 mm/step
CM09	00:22:45	232.90	16.4	83	Tol.: $10^{- 3}$
CM10	00:24:22	267.87	21.8	111	Tol.: $10^{- 3}$
CM11	00:56:20	257.87	21.2	109	Without Line Search

[1] *Corresponding authors

Mesh discretization analysis
Mesh Type	Total nodes	Total elements	Layers on the slab thickness	Element dimension ( *abc* )	Element Type
A	11789	8576	4	25x25x25	HX24L
B	8631	1082	2	50x50x50	CHX60
C	10853	1611	3	50x50x33.3	CHX60
D	45365	8576	4	25x25x25	CHX60
E	54230	14020	5	25x25x20	CHX60
F G H	1330107 19746 23287	28818 3428 4269	6 4 5	16.7x16.7x16.7 Variable Variable	CHX60 CHX60 CHX60

Mesh discretization size analysis
Mesh Type	Layers on the slab thickness	Elapsed time	Peak Load (kN)	$Δ$ P (kN)	Displacement at Peak Load (mm)
B	2	00:22:14	190.35	-	13.6
C	3	00:25:47	214.70	24.35	14.4
D E F	4 5 6	01:32:58 02:20:00 26:27:00	243.45 261.29 268.22	28.75 17.85 6.92	19.8 22.2 23.6

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Aspects of Finite Element Modeling of Punching Shear Behavior of Reinforced Concrete Flat Slabs

Abstract