Distribuição de esforços e confiabilidade de pórticos de concreto armado: íntegro e com remoção de colunas

Facholli, Pedro Henrique Preto; Beck, André Teófilo

Pedro Henrique Preto Facholli

Universidade de São Paulo – USP, Escola de Engenharia de São Carlos, São Carlos, SP, Brasil

http://orcid.org/0000-0002-1933-9472

André Teófilo Beck

Universidade de São Paulo – USP, Escola de Engenharia de São Carlos, São Carlos, SP, Brasil

http://orcid.org/0000-0003-4127-5337

Corresponding author: André Teófilo Beck. E-mail: atbeck@sc.usp.br

Conflict of interest: Nothing to declare.

Author contributions: PF: resources, computer programming, data manipulation, writing original draft; ATB: conceptualization, funding acquisition, methodology, supervision, writing original draft, writing review and editing.

Editors: Mauro Real, Guilherme Aris Parsekian.

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Figure 1
Normal load times bending Moment (NM) interaction diagram for strength of RC columns.

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Figure 2
Degrees of freedom of the frame element [⁴⁵45 L. F. Martha, Análise de Estruturas - Conceitos e Métodos Básicos, 2ª. ed. Editora GEN LTC, 2017, 600 p.].

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Figure 3
Nonlinear behavior in the collapse of a reinforced concrete beam.

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Figure 3
Nonlinear behavior in the collapse of a reinforced concrete beam.

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Figure 4
Studied RC frames and naming convention for columns.

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Figure 5
Illustration of the considered column loss conditions: external (left) and internal (right).

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Figure 6
Reliability indexes for 4-story frame, intact condition.

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Figure 7
Normal load bending moment (NM) interaction diagram for columns, 4-story intact frame.

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Figure 8
Ratio between bending moment and normal force (M/N) for columns at different floors, 4-story intact frame.

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Figure 9
Reliability index and sensitivity coefficients for columns of different floors, 4-story intact frame, external columns (left) and internal columns (right).

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Figure 10
Reliability indexes for 4-story frame, removal of external column, original frame (left) and strengthened frame (right).

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Figure 11
Reliability index and sensitivity coefficients for columns of different floors, strengthened 4-story frame with removal of external column.

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Figure 12
Ratio M/N for columns at different floors, strengthened 4-story frame with removal of external column.

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Figure 13
Reliability indexes for 4-story frame, removal of external column, designed with exceptional load combination of ABNT NBR 8681 [⁴⁸48 Associação Brasileira de Normas Técnicas, Actions and Safety of Structures – Procedure, ABNT NBR 8681, 2003, 18 p.].

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Figure 14
Reliability indexes for 4-story frame, removal of internal column, original frame (left) and strengthened frame (right).

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Figure 15
Reliability index and sensitivity coefficients for columns of different floors, strengthened 4-story frame with removal of internal column.

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Figure 16
Ratio M/N for columns at different floors, strengthened 4-story frame with removal of external column.

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Figure 17
Reliability indexes for the 8-story building for the intact, external column loss and internal column loss, from left to right.

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Table 1
Cross-section dimensions and reinforcement for 4-story building (“CL” means column loss).

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Table 2
Cross-section dimensions and reinforcement for 8-story building (“CL” means column loss).

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Table 3
Nominal values for loads (in kN/m) and ratio $L_{n} / D_{n}$ .

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Table 4
Random variables considered in the reliability analysis and their parameters (“Variable” means “in terms of the standard deviation”).

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Table 5
Collapse probability of the most critical elements, calculated through using $P [H] = 10^{- 5}$ and $P [L D | H] = 1$ (in boldface, values that exceed $10^{- 6}$ , corresponding to $P [C | L D, H] > 0.1$ ).

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Loading case	Width (cm)	Beams		Internal columns		External columns
Loading case	Width (cm)	Height (cm)	Steel reinf. (cm²)	Height (cm)	Steel reinf. (cm²)	Height (cm)	Steel reinf. (cm²)
Normal loading	19.0	50.0	6.80	40.0	5.00	40.0	7.50
Strengthened for external CL	19.0	70.0	16.00	50.0	8.00	50.0	13.00
Strengthened for external CL (ABNT NBR 8681)	19.0	80.0	18.00	50.0	10.00	50.0	17.00
Strengthened for internal CL	19.0	70.0	15.00	50.0	10.00	50.0	14.50

Loading case	Width (cm)	Beams		Internal columns		External columns
Loading case	Width (cm)	Height (cm)	Steel reinf. (cm²)	Height (cm)	Steel reinf. (cm²)	Height (cm)	Steel reinf. (cm²)
Normal loading	19.0	50.0	8.50	60.0	12.00	60.0	6.60
Strengthened for external CL	19.0	80.0	16.00	70.0	12.00	70.0	12.00
Strengthened for internal CL	19.0	80.0	14.00	70.0	20.00	70.0	12.00

Frame	Case	Live load on beams ( $q_{n}$ )	Dead load on beams ( $g_{v, n}$ )	Self-weight of columns ( $g_{p, n}$ )	$L_{n} / D_{n}$ on beams	$L_{n} / D_{n}$ on columns
4-story	Intact	16.00	26.38	1.90	0.61	0.56 - 0.58
4-story	Column loss	16.00	27.33	2.38	0.59	0.52 - 0.56
8-story	Intact	16.00	26.38	2.85	0.61	0.54 - 0.57
8-story	Column loss	16.00	27.80	3.33	0.58	0.51 - 0.56

Variable (symbol)	Distribution	Mean	C.o.V.	Standard deviation	Reference
Cross-section dimensions $(B, H)$	Normal	$d_{n}$	Variable	$4 + 0.006 d_{n}$ (mm)	[⁵⁰50 Joint Committee on Structural Safety, Probabilistic Model Code, 2001. [Online]. Available: https://www.jcss-lc.org/jcss-probabilistic-model-code/ https://www.jcss-lc.org/jcss-probabilist... ]
Self-weight of columns $(G_{p})$	Normal	$1.06 g_{p, n}$	0.12	$0.1272 g_{p, n}$	[³²32 W. C. Santiago, “Reliability-based calibration of partial safety factors from main Brazilian codes,” D.Sc. thesis, São Carlos Sch. Eng., Univ. São Paulo, São Carlos, 2019.]
Dead load on beams $(G_{v})$	Normal	$1.06 g_{v, n}$	0.12	$0.1272 g_{v, n}$	[³²32 W. C. Santiago, “Reliability-based calibration of partial safety factors from main Brazilian codes,” D.Sc. thesis, São Carlos Sch. Eng., Univ. São Paulo, São Carlos, 2019.]
Live load on beams, intact frame $(Q_{50})$	Gumbel	$q_{n}$	0.40	$0.40 q_{n}$	[³²32 W. C. Santiago, “Reliability-based calibration of partial safety factors from main Brazilian codes,” D.Sc. thesis, São Carlos Sch. Eng., Univ. São Paulo, São Carlos, 2019.]
Live load on beams, column loss $(Q_{a . p . t .})$	Gamma	$0.25 q_{n}$	0.55	$0.1375 q_{n}$	[³²32 W. C. Santiago, “Reliability-based calibration of partial safety factors from main Brazilian codes,” D.Sc. thesis, São Carlos Sch. Eng., Univ. São Paulo, São Carlos, 2019.]
Concrete compressive strength $(f_{c})$	Normal	$1.22 f_{c k}$	0.15	$0.183 f_{c k}$	[³²32 W. C. Santiago, “Reliability-based calibration of partial safety factors from main Brazilian codes,” D.Sc. thesis, São Carlos Sch. Eng., Univ. São Paulo, São Carlos, 2019.]
Steel yield strength $(f_{y})$	Normal	$1.22 f_{y k}$	0.04	$0.0488 f_{y k}$	[³²32 W. C. Santiago, “Reliability-based calibration of partial safety factors from main Brazilian codes,” D.Sc. thesis, São Carlos Sch. Eng., Univ. São Paulo, São Carlos, 2019.]
Model error for beam bending $(E_{B})$	Lognormal	0.99	0.024	0.02376	[¹⁹19 E. S. Santos, “Avaliação estatística do erro de modelos de resistência para elementos lineares de concreto armado da ABNT NBR 6118:2007,” in M.S. thesis, São Carlos Sch. Eng., Univ. São Paulo, São Carlos, 2012.]
Model error for columns $(E_{C})$	Normal	1.15	0.145	0.16675	[⁴⁹49 W. C. Santiago, H. M. Kroetz, S. H. C. Santos, F. R. Stucchi, and A. T. Beck, "Reliability-based calibration of main brazilian structural design codes," Lat. Am. J. Solids Struct., vol. 17, no. 1, pp. 1–28, 2020, http://dx.doi.org/10.1590/1679-78255754. http://dx.doi.org/10.1590/1679-78255754... ]

Structure		Beam	External Column	Internal Column
4-story	Intact	3.36E-03	1.31E-03	3.17E-05
	ECL - O	1.00E-05	9.63E-06	9.96E-06
	ECL - S	2.80E-08	1.46E-07	1.19E-07
	ECR - S (ABNT NBR 8681)	8.16E-11	2.70E-09	4.07E-10
	ICL - O	1.00E-05	9.98E-06	7.91E-06
	ICL - S	9.35E-09	9.14E-08	4.07E-10
8-story	Intact	6.57E-03	5.39E-03	2.70E-04
	ECL - O	1.00E-05	9.79E-06	1.66E-07
	ECL - S	1.66E-06	1.64E-06	1.69E-08
	ICL - O	1.00E-05	9.73E-06	1.44E-08
	ICL - S	1.22E-08	1.99E-08	3.49E-09

[1] Corresponding author: André Teófilo Beck. E-mail: atbeck@sc.usp.br

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Distribuição de esforços e confiabilidade de pórticos de concreto armado: íntegro e com remoção de colunas

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