Analysis of the global second-order effects on irregular reinforced concrete structures using the natural period of vibration

LEITÃO, F. F.; SIQUEIRA, G. H.; VIEIRA JR., L. C. M.; ALMEIDA, S. J. C.

doi:10.1590/S1983-41952019000200012

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Articles • Rev. IBRACON Estrut. Mater. 12 (02) • Mar-Apr 2019 • https://doi.org/10.1590/S1983-41952019000200012 copy

Analysis of the global second-order effects on irregular reinforced concrete structures using the natural period of vibration

Análise dos efeitos globais de segunda ordem em estruturas irregulares em concreto armado utilizando o período natural de vibração

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The χ_T parameter, a simplified method recently presented, allows to estimate the global second-order effects on reinforced concrete frames using the natural period of vibration. This parameter was developed based on the fact that both natural period of vibration and global second-order effects depend essentially on the stiffness and mass matrices of the structure, being thus related. In this paper, numerical analyses are conducted on nine models with different patterns of irregularity in terms of geometry in plan and stiffness. The main purpose of these analyses is to evaluate the applicability of the χ_T parameter in asymmetric structures as well as that can present torsional modes as the fundamental mode of vibration. In addition, different hypotheses are tested in order to verify the influence of the different modes of vibration in the structural sensitivity to global second-order effects. Results of the simplified analyses were compared to the final bending moment values obtained through a nonlinear numerical analysis considering the P-Δ effect. It is observed that the parameter χ_T is a promising indicator for a simplified estimation of the global second-order effects for concrete frames, especially when higher modes of vibration are taken account in the analysis.

Keywords:
global second-order effects; natural period of vibration; modal analysis

IBRACON - Instituto Brasileiro do Concreto Instituto Brasileiro do Concreto (IBRACON), Av. Queiroz Filho, nº 1700 sala 407/408 Torre D, Villa Lobos Office Park, CEP 05319-000, São Paulo, SP - Brasil, Tel. (55 11) 3735-0202, Fax: (55 11) 3733-2190 - São Paulo - SP - Brazil
E-mail: arlene@ibracon.org.br

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Model	Maximum number of floors	Total height (m)	Characteristic compressive strength (MPa)	Modulus of elasticity (MPa)	Vertical loading per floor (kN/floor)	γ_z (x)	γ_z (y)
1	22	66.0	25	23,800	5,650	1.16	1.32
2	21	63.0	25	23,800	3,720	1.18	1.29
3	28	84.0	25	23,800	3,220	1.20	1.31
4	43	129.0	25	23,800	8,690	1.20	1.18
5	27	81.0	35	28,161	13,460	1.16	1.27
6	30	90.0	35	28,161	4,570	1.16	1.31
7	30	90.0	35	28,161	5,070 (typical floor)	1.17	1.26
8	33	66.0	35	28,161	4,580 (typical floor)	1.15	1.11
9	21	63.0	25	23,800	3,720	1.19	1.27

Hypothesis	Description
I	1^st flexural mode in the considered direction - original hypothesis (REIS et al. [⁸[8] WILSON, E. L.; EERI, M. e HABIBULLAH, A.. Static and Dynamic Analysis of Multi-Story Buildings Including P-Delta Effects. Earthquake Spectra, Vol. 3(2), pp. 289-298, 1987.])
II	1^st mode of vibration (fundamental mode)
III	Average of the natural period of vibration weighted by effective modal mass

Mode of vibration	Period (s)	Modal participation factor (%)
Mode of vibration	Period (s)	X-direction	Y-direction	Accumulated sum X-direction	Accumulated sum Y-direction	Z-rotation
1	7.09	0.09 %	72.38 %	0.09 %	72.38 %	42.75 %
2	6.38	0.74 %	0.67 %	0.83 %	73.05 %	8.74 %
3	5.70	75.58 %	0.05 %	76.41 %	73.10 %	23.66 %
4	2.14	0.05 %	8.81 %	76.46 %	81.91 %	7.50 %
5	2.07	0.05 %	2.70 %	76.51 %	84.61 %	0.01 %
6	1.83	10.30 %	0.01 %	86.81 %	84.62 %	3.22 %
7	1.21	0.03 %	0.19 %	86.84 %	84.81 %	1.14 %
8	1.08	0.01 %	4.24 %	86.85 %	89.05 %	1.64 %
9	1.02	3.60 %	0.02 %	90.45 %	89.07 %	1.22 %
10	0.82	0.01 %	0.05 %	90.46 %	89.12 %	0.52 %
11	0.69	1.96 %	0.01 %	92.42 %	89.13 %	0.45 %
12	0.67	0.01 %	2.45 %	92.43 %	91.58 %	1.24 %

Model	Direction	10-Storey		20-Storey			Maximum number of storey
Model	Direction	75 %	90 %	75 %	90 %	75 %	90 %
1	X	1,027	1,028	1,069	1,077	1,079	1,089
1	Y	1,122	1,036	1,112	1,132	1,153	1,158
2	X	1,052	1,057	1,147	1,166	1,159	1,179
2	Y	1,066	1,068	1,235	1,252	1,261	1,277
3	X	1,044	1,049	1,124	1,138	1,205	1,230
3	Y	1,035	1,035	1,146	1,150	1,302	1,303
4	X	1,044	1,045	1,117	1,131	1,428	1,453
4	Y	1,023	1,023	1,086	1,088	1,378	1,393
5	X	1,024	1,025	1,100	1,103	1,046	1,047
5	Y	1,046	1,047	1,179	1,185	1,082	1,084
6	X	1,014	1,015	1,055	1,057	1,119	1,123
6	Y	1,035	1,038	1,121	1,128	1,251	1,265
7	X	1,010	1,011	1,031	1,033	1,082	1,083
7	Y	1,019	1,020	1,058	1,061	1,172	1,177
8	X	1,016	1,017	1,047	1,048	1,108	1,113
8	Y	1,015	1,015	1,038	1,039	1,082	1,086
9	X	1,067	1,069	1,197	1,204	1,207	1,215
9	Y	1,070	1,072	1,258	1,268	1,256	1,287

Legend	Name	Description
	χ_{T(I - 1} ^st _{Flexural Mode Reis et al. (2017))}	Hypothesis I (1^st flexural mode)
	χ_{T(II - 1} ^st _{Mode of Vibration)}	Hypothesis II (fundamental mode of vibration)
	χ_{T(III - Weighted Average)}	Hypothesis III (weighted average)
	Columns	Columns Amplification M₂/M₁
	γ_Z	Coefficient γ_Z values
	Critical columns	Critical columns amplification M₂/M₁

Models	1^st mode of vibration (fundamental)
Models	X-Flexural	Y-Flexural	Torsional	Y-Flexural/Torsional
1	-	15-22 Storey	1-14 Storey	-
2	-	12-21 Storey	1-11 Storey	-
3	1-14 Storey	-	-	14-28 Storey
4	1-43 Storey	-	-	-
5	-	24-27 Storey	1-23 Storey	-
6	-	-	1-18 Storey	19-30 Storey
7	-	9-30 Storey	3-8 Storey	-
8	8-33 Storey	-	3-7 Storey	-
9	-	14-21 Storey	1-13 Storey	-

Model	Model of vibration	Natural period of vibration (s)	Mode of vibration type
Model 1 (14-Storey)	1º	3.39	Torsional (rotation along z-axis)
	2º	3.38	Flexural in y-direction
	3º	2.64	Flexural in x-direction
Model 2 (10-Storey)	1º	2.95	Torsional (rotation along z-axis)
	2º	2.84	Flexural in y-direction
	3º	2.50	Flexural in x-direction
Model 5 (23-Storey)	1º	7.47	Torsional (rotation along z-axis)
	2º	7.04	Flexural in y-direction
	3º	5.73	Flexural in x-direction
Model 9 (10-Storey)	1º	3.71	Torsional (rotation along z-axis)
	2º	3.44	Flexural in x-direction
	3º	3.34	Flexural in y-direction

Number of Storey	Column	Base cross section (cm x cm)	Moment of Inertia in x-direction (cm⁴)	1^st order base moment - M1 (kN.m)	Final base moment - M2 (kN.m)	Amplification M2/M1	Critical column
12-Storey	P9	20 x 88	597226.68	124.6	144.9	1.16	X
12-Storey	P17	50 x 50	520833.33	108	123.5	1.14	-
13-Storey	P9	25 x 88	767158.33	136.3	156.5	1.15	-
13-Storey	P17	45 x 65	1029843.75	180.8	205.5	1.14	X