Comportamento pós punção e colapso progressivo em uma estrutura em laje lisa

Galdino, Bernardo Cruz Pereira; Melo, Guilherme Sales

Bernardo Cruz Pereira Galdino

overall responsibility

Universidade de Brasília – UnB, Department of Civil and Environmental Engineering, Brasília, DF, Brasil

https://orcid.org/0000-0002-5644-4570

Guilherme Sales Melo

supervision

Universidade de Brasília – UnB, Department of Civil and Environmental Engineering, Brasília, DF, Brasil

https://orcid.org/0000-0001-9417-9010

Corresponding author: Bernardo Cruz Pereira Galdino. E-mail: eng.bernardogaldino@gmail.com

Conflict of interest: Nothing to declare.

Author contributions: The second author of this article was the supervisor of the research accomplished by the first author during his MSc Thesis.

Editors: Diogo Rodrigo Ribeiro, Guilherme Aris Parsekian.

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Figure 1
Floor plan of the structure – Dimensions in m.

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Figure 2
Acting and strength stresses ratio in the support.

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Figure 3
Comparison of reinforcement steel area.

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Figure 4
Variation of support reactions after failure at P2.

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Figure 5
Shear stresses ratio in P1 for zero and for 60% of residual reaction in P2.

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Figure 6
Positive and negative yield line configuration.

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Figure 7
Deformation of a slab – Dimensions in cm.

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Figure 8
Positive moment strengths in the X direction – Dimensions in m.

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Figure 9
Positive moment strengths in the Y direction – Dimensions in m.

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Figure 10
Negative moment strengths in the X direction – Dimensions in m.

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Figure 11
Negative moment strengths in the Y direction – Dimensions in m.

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Figure 2
Acting and strength stresses ratio in the support.

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Figure 3
Comparison of reinforcement steel area.

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Figure 4
Variation of support reactions after failure at P2.

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Figure 5
Shear stresses ratio in P1 for zero and for 60% of residual reaction in P2.

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Figure 6
Positive and negative yield line configuration.

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Figure 7
Deformation of a slab – Dimensions in cm.

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Figure 8
Positive moment strengths in the X direction – Dimensions in m.

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Figure 9
Positive moment strengths in the Y direction – Dimensions in m.

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Figure 10
Negative moment strengths in the X direction – Dimensions in m.

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Figure 11
Negative moment strengths in the Y direction – Dimensions in m.

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Figure 12
Section of the waffle slab – Dimensions in cm.

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Figure 13
Floor plan of the parking – Dimensions in cm.

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Figure 14
Acting and strength stresses ratio in support.

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Figure 15
Comparison of reinforcement steel areas.

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Figure 16
Variation of support reactions after failure at P23.

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Figure 17
Shear stresses ratio in P27 for zero and for 60% of residual reaction in P23.

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Figure 18
Positive and negative yield line configuration.

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Figure 19
Deformations of slab – Dimensions in cm.

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Figure 20
Positive moment strengths in the X direction – Dimensions in cm.

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Figure 21
Negative moment strengths in the X direction – Dimensions in cm.

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Figure 22
Negative moment strengths in the Y direction – Dimensions in cm.

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Figure 14
Acting and strength stresses ratio in support.

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Figure 15
Comparison of reinforcement steel areas.

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Figure 16
Variation of support reactions after failure at P23.

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Figure 17
Shear stresses ratio in P27 for zero and for 60% of residual reaction in P23.

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Figure 18
Positive and negative yield line configuration.

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Figure 19
Deformations of slab – Dimensions in cm.

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Figure 20
Positive moment strengths in the X direction – Dimensions in cm.

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Figure 21
Negative moment strengths in the X direction – Dimensions in cm.

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Figure 22
Negative moment strengths in the Y direction – Dimensions in cm.

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Table 1
Load used.

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Table 2
Reactions and the moments of the support in the structure without damage.

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Table 3
Shear stresses at the critical section.

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Table 4
Integrity reinforcement area.

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Table 5
Reactions and bending moments of support after P2 punching shear.

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Table 6
Shear stresses in P1 after failure at P2.

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Table 7
Load used.

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Table 8
Reactions and the moments of the support in the structure without damage.

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Table 9
Shear stresses at the critical section.

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Table 10
Integrity reinforcement area.

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Table 11
Reactions and moments of support after P23 punching shear.

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Table 12
Shear stresses in P27 after failure at P23.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Overview	Amount
Dead load (distributed):	Load (kN/m²)
Self-weight of slab (h = 16.0 cm)	25.0 $∙$ 0.16 = 4.00
Cladding	0.80
Wall (middle of the slab)	1.00
Wall (periphery of the slab)	2.28
Live load:	Load (kN/m²)
Load in residential building	1.50

Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)	Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)
P1	177.10	21.80	23.00	P7	290.50	0.00	4.20
P2	228.60	4.50	18.10	P8	239.30	20.50	0.00
P3	228.60	4.50	18.10	P9	177.10	21.80	23.00
P4	177.10	21.80	23.00	P10	228.60	4.50	18.10
P5	239.30	20.50	0.00	P11	228.60	4.50	18.10
P6	290.50	0.00	4.20	P12	177.10	21.80	23.00

Codes	Stress (MPa)	P1	P2	P6
	$τ_{r d 3}$	0.70	0.74	0.78
	$τ_{s d 3}$	0.48	0.48	0.57
NBR 6118	$τ_{r d 2}$	1.53	1.56	1.60
	$τ_{s d 2}$	1.03	1.06	1.17
	$τ_{r d 1}$	5.09	5.09	5.09
	$τ_{s d 1}$	3.73	3.15	2.86
	$σ_{r d 3}$	0.58	0.61	0.64
	$σ_{s d 3}$	0.52	0.51	0.61
EUROCODE	$σ_{r d 2}$	1.42	1.45	1.48
	$σ_{s d 2}$	1.03	1.06	1.17
	$σ_{r d 1}$	5.28	5.28	5.28
	$σ_{s d 1}$	3.73	3.15	2.86
	$v_{c o u t}$	0.70	0.70	0.70
ACI	$V_{u 2}$	0.43	0.43	0.52
	$v_{n}$	2.17	2.17	2.17
	$V_{u 1}$	1.34	1.31	1.38

Column	$N_{k}$ (kN)	NBR 6118 (cm²)	GSA (cm²)	CEB (cm²)	ACI (cm²)
P1	177.10	7.33	10.81	8.23	6.46
P2	228.60	9.46	10.81	10.62	6.46
P6	290.50	12.03	10.81	13.50	6.46

Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)	Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)
P1	271.00	127.10	17.30	P7	267.80	7.30	1.90
P2	-	-	-	P8	235.70	19.00	0.90
P3	349.40	74.80	18.90	P9	172.10	11.30	20.90
P4	158.20	20.50	13.90	P10	207.90	4.30	5.70
P5	208.20	30.90	13.20	P11	223.40	4.50	12.20
P6	411.20	10.90	42.40	P12	178.90	22.60	23.70

Codes	Stress (MPa)	P1 No residual reaction in P2	P1 60% of residual reaction in P2
	$τ_{r d 3}$	0.70	0.70
	$τ_{s d 3}$	0.93	0.66
NBR 6118	$τ_{r d 2}$	1.53	1.53
	$τ_{s d 2}$	2.19	1.50
	$τ_{r d 1}$	5.09	5.09
	$τ_{s d 1}$	9.34	6.02
	$σ_{r d 3}$	0.58	0.58
	$σ_{s d 3}$	0.98	0.70
EUROCODE	$σ_{r d 2}$	1.42	1.42
	$σ_{s d 2}$	2.19	1.50
	$σ_{r d 1}$	5.28	5.28
	$σ_{s d 1}$	9.34	6.02
	$v_{c o u t}$	0.70	0.70
ACI	$V_{u 2}$	0.79	0.58
	$v_{n}$	2.17	2.17
	$V_{u 1}$	2.97	2.00

Overview	Amount
Dead Load (distributed):	Load (kN/m²)
Self-weight of slab	4.60
Cladding	1.23
Wall	0.19
Live Load:	Load (kN/m²)
Load in parking building	5.00

Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)	Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)
P1	117.30	9.50	28.30	P18	465.70	3.50	30.10
P2	99.60	18.10	4.00	P19	220.40	0.30	14.10
P3	134.90	15.50	44.40	P20	82.90	5.40	41.30
P4	156.80	31.50	18.80	P21	128.10	7.50	35.90
P5	215.00	34.30	9.50	P22	193.00	10.20	39.10
P6	145.90	11.40	26.90	P23	415.70	1.30	57.30
P7	546.40	83.60	23.50	P24	163.40	11.30	65.20
P8	500.70	33.50	20.50	P25	240.30	13.60	102.60
P9	332.90	6.00	43.30	P26	317.90	9.00	29.20
P10	159.70	39.60	40.90	P27	530.00	25.70	107.30
P11	332.10	20.20	15.70	P28	424.30	3.40	33.90
P12	56.90	1.90	6.20	P29	186.40	4.40	36.20
P13	86.30	4.60	28.30	P30	153.60	27.80	40.90
P14	243.40	31.90	54.20	P31	278.30	59.40	0.60
P15	373.90	75.50	31.80	P32	255.40	48.40	1.40
P16	590.70	20.50	69.10	P33	131.00	35.30	19.70
P17	173.10	9.10	45.60	-	-	-	-

Codes	Stress (MPa)	P11	P23	P27
	$τ_{r d 3}$	0.52	0.51	0.56
	$τ_{s d 3}$	0.15	0.19	0.24
NBR 6118	$τ_{r d 2}$	0.61	0.61	0.65
	$τ_{s d 2}$	0.31	0.40	0.52
	$τ_{r d 1}$	5.09	5.09	5.09
	$τ_{s d 1}$	1.29	2.00	1.99
	$σ_{r d 3}$	0.48	0.48	0.52
	$σ_{s d 3}$	0.16	0.20	0.26
EUROCODE	$σ_{r d 2}$	0.52	0.52	0.55
	$σ_{s d 2}$	0.31	0.40	0.52
	$σ_{r d 1}$	5.28	5.28	5.28
	$σ_{s d 1}$	1.29	2.00	1.99
	$v_{c o u t}$	0.66	0.66	0.66
ACI	$V_{u 2}$	0.14	0.17	0.22
	$v_{n}$	1.17	1.18	1.17
	$V_{u 1}$	0.43	0.58	0.69

Column	$N_{k}$ (kN)	NBR 6118 (cm²)	GSA (cm²)	CEB (cm²)	ACI (cm²)
P11	332.10	13.75	25.55	15.43	5.62
P23	415.70	17.21	25.55	19.32	4.67
P27	530.00	21.94	25.55	24.63	5.77

Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)	Column	$N_{k}$ (kN)	$M_{x}$ (kNm)	$M_{y}$ (kNm)
P1	116.70	9.30	27.90	P17	182.30	7.20	50.10
P2	99.60	18.10	3.30	P18	520.00	3.20	27.60
P3	134.60	15.50	43.6	P19	210.10	0.20	18.20
P4	158.50	32.20	19.4	P20	82.40	5.70	46.60
P5	217.40	33.80	10.50	P21	117.10	7.90	38.10
P6	145.40	10.80	27.3	P22	220.10	8.50	35.10
P7	545.00	82.90	23.3	P23	-	-	-
P8	486.40	30.00	22.60	P24	164.50	12.10	63.50
P9	325.00	7.60	40.40	P25	242.60	16.00	99.70
P10	160.50	39.30	43.40	P26	334.60	10.40	36.80
P11	325.70	20.30	12.10	P27	714.30	34.70	105.40
P12	58.40	1.90	5.90	P28	415.00	1.90	41.00
P13	85.20	4.70	26.70	P29	185.00	4.20	38.30
P14	242.30	32.00	59.80	P30	155.40	28.10	43.00
P15	379.40	74.60	30.60	P31	273.40	54.20	0.50
P16	752.10	25.20	158.6	P32	245.60	55.70	1.20
-	-	-	-	P33	130.90	34.50	22.20

Codes	Stress (MPa)	P27 No residual reaction in P23
	$τ_{r d 3}$	0.56	0.56
	$τ_{s d 3}$	0.32	0.27
NBR 6118	$τ_{r d 2}$	0.65	0.65
	$τ_{s d 2}$	0.67	0.58
	$τ_{r d 1}$	5.09	5.09
	$τ_{s d 1}$	2.45	2.18
	$σ_{r d 3}$	0.52	0.52
	$σ_{s d 3}$	0.34	0.29
EUROCODE	$σ_{r d 2}$	0.55	0.55
	$σ_{s d 2}$	0.66	0.58
	$σ_{r d 1}$	5.28	5.28
	$σ_{s d 1}$	2.45	2.18
	$v_{c o u t}$	0.66	0.66
ACI	$V_{u 2}$	0.29	0.25
	$v_{n}$	1.17	1.17
	$V_{u 1}$	0.88	0.77

[1] Corresponding author: Bernardo Cruz Pereira Galdino. E-mail: eng.bernardogaldino@gmail.com

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Comportamento pós punção e colapso progressivo em uma estrutura em laje lisa

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