Análise dinâmica de uma laje aliviada biaxial submetida a ações humanas

Mota, Tiago; Oliveira, Marcos Honorato; Doz, Graciela

Tiago Mota

conceptualization

methodology

formal analysis

original draft preparation

writing

review

editing

Universidade de Brasília – UnB, Departamento de Engenharia Civil e Ambiental, Brasília, DF, Brasil

http://orcid.org/0000-0002-7682-0473

Marcos Honorato Oliveira

reviewing

Universidade de Brasília – UnB, Departamento de Engenharia Civil e Ambiental, Brasília, DF, Brasil

http://orcid.org/0000-0003-1251-7923

Graciela Doz

conceptualization

formal analysis

data curation

supervision

reviewing

Universidade de Brasília – UnB, Departamento de Engenharia Civil e Ambiental, Brasília, DF, Brasil

http://orcid.org/0000-0001-6428-4012

Corresponding author: Tiago Mota. E-mail: tiagomoota@gmail.com

Conflict of interest: Nothing to declare.

Editors: Túlio Nogueira Bittencourt, Daniel Carlos Taissum Cardoso.

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Figure 1
Floor plan of the analyzed floor.

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Figure 2
Detail of a generic section of the analyzed pavement, in centimeters.

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Figure 3
(a) Model 1 – Linear Element Frame. (b) Model 2 – Element plan Shell-Thin. (c) Model 3 – Element plan Shell-Layered. (d) Model 4 – Unit of a solid element.

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Figure 4
Cross section of the shell-layered slab, in centimeters.

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Figure 5
Three-dimensional representation of the studied rigid pavement model, using a Shell Layered finite element.

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Figure 6
Location of the nodes analyzed in this work (dimensions in cm).

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Figure 3
(a) Model 1 – Linear Element Frame. (b) Model 2 – Element plan Shell-Thin. (c) Model 3 – Element plan Shell-Layered. (d) Model 4 – Unit of a solid element.

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Figure 4
Cross section of the shell-layered slab, in centimeters.

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Figure 5
Three-dimensional representation of the studied rigid pavement model, using a Shell Layered finite element.

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Figure 6
Location of the nodes analyzed in this work (dimensions in cm).

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Figure 7
Representation of the first three vibration modes of the analyzed pavement.

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Figure 8
Acceleration response obtained for the 4 nodes analyzed for the jumping action.

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Table 1
Fourier coefficients for dynamic actions [¹⁴14 H. Bachmann, Vibration Problems in Structures: Practical Guidelines. Basel: Birkhӓuser Verlag, 1995.].

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Table 2
Acceptable levels of vibrations for various uses [¹⁵15 H. Bachmann and W. Ammann, Vibrations in Structures Induced by Man and Machines. Zurich: IABSE, 1987.].

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Table 3
Value ranges for vertical accelerations Sétra /AFGG [¹⁶16 SÉTRA, Technical Guide - Footbridges - Assessment of Vibrational Behaviour of Footbridges Under Pedestrian Loading, 2006.].

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Table 4
Recommended tolerance acceleration limits for rhythmic activities in buildings [¹⁸18 American Institute of Steel Construction, Vibrations of Steel-Framed Structural Systems Due to Human Activity, 2016, p. 138.].

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Table 5
Physical properties of the materials used.

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Table 6
Natural frequencies obtained experimentally and with numerical analysis.

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Table 7
Damping coefficient [¹⁴14 H. Bachmann, Vibration Problems in Structures: Practical Guidelines. Basel: Birkhӓuser Verlag, 1995.]

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Table 8
Modal characteristics of the analyzed pavement.

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Table 9
Maximum dynamic responses to skipping loading.

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Table 13
Maximum dynamic responses to loading from hand clapping with body bouncing while standing.

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Table 10
Maximum dynamic responses to dancing loading.

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Table 11
Maximum dynamic responses to loading from clapping.

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Table 12
Maximum dynamic responses to loading from body swaying.

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Table 14
Dynamic responses from jumping and dancing activities.

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Representative type of activity	Activity rate (Hz)		Fourier coefficient and phase angle					Density of the project (People/m²)
Representative type of activity	Activity rate (Hz)		α ₁	α ₂	Φ ₂	α ₃	Φ ₃	Density of the project (People/m²)
Jumping	Normal	2	1.8	1.3	*)	0.7	*)	Fitness training~ 0.25 (in extreme cases up to 0.5)
	Normal	3	1.7	1.1	*)	0.5	*)	Fitness training~ 0.25 (in extreme cases up to 0.5)
	High	2	1.9	1.6	*)	1.1	*)	) Φ₂ = Φ₃ = π(1-f_pT_p*)
	High	3	1.8	1.3	*)	0.8	*)	) Φ₂ = Φ₃ = π(1-f_pT_p*)
Dancing		2 to 3	0.5	0.15		0.1		~ 4 (in extreme cases up to 6)
Hand clapping with body bouncing while standing		1.6	0.17	0.1		0.04		No fixed seating ~ 4 (in extreme cases up to 6)
Hand clapping with body bouncing while standing		2.4	0.38	0.12		0.02		Fixed seating ~ 2 to 3
Hand Clapping	Normal	1.6	0.024	0.01		0.009		~ 2 to 3
		2.4	0.047	0.024		0.015
	Intensive	2	0.017	0.047		0.037
Lateral body swaying	Seated	0.6	α₁/2 = 0.4	-		-		~ 3 to 4
Lateral body swaying	Standing	0.6	α₁/2 = 0,5	-		-		~ 3 to 4

Structure	Acceptable levels	Comments
Pedestrian structures	a ≤ 5 to 10% of g	Lower values usually do not produce discomfort.
Commercial buildings	a ≤ 2% de g	DIN 4150 and BS 6472 have values different from these
Gyms and sport	a ≤ 5 to 10% of g
Dance and concert halls	a ≤ 5 to 10% of g	The same for the gym
Industrial floors	v ≤ ± 10mm/s	For conventional industries; high-intensity industries need tighter limits

Acceleration levels	0(m/s²)	0.5 (m/s²)	1 (m/s²)	2.5 (m/s²)
Level 1	Maximum
Level 2		Medium
Level 3			Minimum
Level 4				Unacceptable

Affected Occupancy	Tolerance Acceleration Limit, a, %g
Office or residential	0.5
Dining	1.5-2.5
Weightlifting	1.5-2.5
Rhythmic activity only	4-7

Physical properties	Concrete	HDPE (high-density polyethylene)
Compression strength (MPa)	30	20
Elastic Modulus (GPa)	24	0.827
Density (kN/m³)	24	11.67

Modes	Experimental Result	Model 1	Variation (%)	Model 2	Variation (%)	Model 3	Variation (%)	Model 4	Variation (%)
Mode 1	6.7	9.58	43	6.79	1.32	6.74	0.6	6.83	1.94
Mode 2	12.6	21.16	68	13.86	10	13.55	7.54	14.05	11.50
Mode 3	18.38	31.43	71	20.39	10.93	19.60	6.63	19.98	8.70

MATERIAL	ζ
Non-cracked reinforced concrete with low-stress level	0.007 to 0.010
Reinforced concrete completely cracked, with a medium stress level	0.010 to 0.040
Completely cracked reinforced concrete, with a high level of tension, but without steel yielding.	0.005 to 0.008
Non-cracked prestressed concrete	0.004 to 0.007
Concrete with limited prestressing, slightly cracked	0.008 to 0.012
Composite materials	0.002 to 0.003
Steel	0.001 to 0.002

Vibration modes	Natural frequency (f)	Period
Mode 1	2.37 Hz	0.42 s
Mode 2	2.77 Hz	0.36 s
Mode 3	3.29 Hz	0.30 s

Jumping
Node	Displacement (m)	Velocity (m/s)	Acceleration (m/s²)
1	-0.00177	0.0305	0.55
2	-0.00306	0.0456	0.85
3	-0.0038	0.0598	1.16
4	-0.00456	0.0764	1.26

Hand clapping with body bouncing while standing
Node	Displacement (m)	Velocity (m/s)	Acceleration (m/s²)
1	0.00026	0.0025	0.04
2	0.00125	0.0088	0.134
3	0.00159	0.011	0.172
4	0.00176	0.012	0.185

Dancing
Node	Displacement (m)	Velocity (m/s)	Acceleration (m/s²)
1	-0.0008	0.011	0.17
2	-0.00238	0.027	0.44
3	-0.00311	0.036	0.59
4	-0.00387	0.0465	0.75

Clapping Hands
Node	Displacement (m)	Velocity (m/s)	Acceleration (m/s²)
1	0.00026	0.0025	0.04
2	0.00125	0.0088	0.134
3	0.00159	0.011	0.172
4	0.00176	0.012	0.185

Body swaying
Node	Displacement (m)	Velocity (m/s)	Acceleration (m/s²)
1	-0.085	0.025	0.15
2	-0.0111	0.028	0.21
3	-0.0116	0.03	0.31
4	-0.0125	0.037	0.32

Activity	Node	Maximum Acceleration (m/s²)	Bachman Limit (m/s²)	Comfort level Sétra/AFGG	Steel Design Guide 11
14Jumping	1	0.55	Out of limit	Between medium and minimum	within the limit
	2	0.85		Between medium and minimum	Out of limit
	3	1.16		Between minimal and unacceptable	Out of limit
	4	1.26		Between minimal and unacceptable	Out of limit
Dancing	1	0.17	within the limit	Between maximum and average.	within the limit
	2	0.44	within the limit	Between maximum and average.	within the limit
	3	0.59	out of limit	Between medium and minimum	within the limit
	4	0.75	out of limit	Between medium and minimum	Out of limit

[1] Corresponding author: Tiago Mota. E-mail: tiagomoota@gmail.com

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Análise dinâmica de uma laje aliviada biaxial submetida a ações humanas

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