Punching shear design with control perimeters subjected to asymmetrical bending

Souza, Mariana Nunes; Perlingeiro, Mayra Soares Pereira Lima; Schulz, Mauro

doi:10.1590/S1983-41952023000300007

Mariana Nunes Souza

literature review

numerical tests

validation

writing

proofreading

Universidade Federal Fluminense – UFF, Civil Engineering Department, Niteroi, RJ, Brasil

https://orcid.org/0000-0002-2849-9742

Mayra Soares Pereira Lima Perlingeiro

conceptualization

literature review

methodology

writing

proofreading

supervision

Universidade Federal Fluminense – UFF, Civil Engineering Department, Niteroi, RJ, Brasil

https://orcid.org/0000-0002-2807-0057

Mauro Schulz

conceptualization

literature review

methodology

formal analysis

computer programming

validation

writing

proofreading

supervision

Universidade Federal Fluminense – UFF, Civil Engineering Department, Niteroi, RJ, Brasil

https://orcid.org/0000-0003-3987-3143

Corresponding author: Mauro Schulz. E-mail: mschulz@id.uff.br

Conflict of interest: Nothing to declare.

Author contributions: MNS: literature review, numerical tests, validation, writing, proofreading; MSPLP: conceptualization, literature review, methodology, writing, proofreading, supervision; MS: conceptualization, literature review, methodology, formal analysis, computer programming, validation, writing, proofreading, supervision.

Editors: Leandro Trautwein, Mauro Real, Mario Pimentel.

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Figure 1
Unbalanced moment transfer in slab-column connections

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Figure 2
Coordinate system, applied forces and moments, and distributed shear forces

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Figure 3
Rotated coordinate system associated with the principal moments

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Figure 4
Discretization of control perimeters into segments

S_{i}

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Figure 5
Parameter

s

and unit shear force distribution along the control perimeter

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Figure 6
Partial integration between points

P

and

Q

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Figure 7
Unit shear force along the control perimeter

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Figure 4
Discretization of control perimeters into segments

S_{i}

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Figure 5
Parameter

s

and unit shear force distribution along the control perimeter

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Figure 6
Partial integration between points

P

and

Q

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Figure 7
Unit shear force along the control perimeter

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Figure 8
Examples: asymmetrical diagrams of unit shear forces

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Table 1
Integration of segment

I J

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Table 2
Combinations of criteria from Model Code 90, Eurocode 2, and NBR 6118

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Table 4
C5 results for experiments in Ferreira [¹⁹19 M. P. Ferreira, "Punção em lajes lisas de concreto armado com armaduras de cisalhamento e momentos desbalanceados", Ph.D. dissertation, Fac. Tecnol., Univ. Brasília, Brasília, 2010. [Online]. Available: https://repositorio.unb.br/handle/10482/8965
https://repositorio.unb.br/handle/10482/... ] and Ferreira et al. [²⁰20 M. P. Ferreira, G. S. S. A. Melo, P. E. Regan, and R. L. Vollum, "Punching of reinforced concrete flat slabs with double-headed shear reinforcement," ACI Struct. J., vol. 111, no. 2, pp. 363–374, Jan. 2014, http://dx.doi.org/10.14359/51686535.
http://dx.doi.org/10.14359/51686535... ].

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Table 5
C5 results for experiments in Ferreira [¹⁹19 M. P. Ferreira, "Punção em lajes lisas de concreto armado com armaduras de cisalhamento e momentos desbalanceados", Ph.D. dissertation, Fac. Tecnol., Univ. Brasília, Brasília, 2010. [Online]. Available: https://repositorio.unb.br/handle/10482/8965
https://repositorio.unb.br/handle/10482/... ] and Ferreira et al. [²¹21 M. P. Ferreira, M. H. Oliveira, and G. S. S. A. Melo, "Tests on the punching resistance of flat slabs with unbalanced moments," Eng. Struct., vol. 196, no. July, 2019, http://dx.doi.org/10.1016/j.engstruct.2019.109311.
http://dx.doi.org/10.1016/j.engstruct.20... ].

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Table 6
C5 results for experiments in Feliciano [²²22 F. M. H. Feliciano, "Punção em lajes lisas de concreto armado com pilares de borda e excentricidade externa", M.S. thesis, Fac. Tecnol., Univ. Brasília, Brasília, 2011. [Online]. Available: https://repositorio.unb.br/handle/10482/9354
https://repositorio.unb.br/handle/10482/... ].

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Table 7
C5 results for experiments in Barbosa [²³23 W. C. S. Barbosa, "Punção em lajes lisas de concreto armado com pilares de canto reentrante", M.S. thesis, Fac. Tecnol., Univ. Brasília, Brasília, 2012. [Online]. Available: https://repositorio.unb.br/handle/10482/10968
https://repositorio.unb.br/handle/10482/... ].

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Table 8
C5 results for experiments in Oliveira [²⁴24 M. H. Oliveira, "Punção em lajes lisas com armadura de cisalhamento submetidas a carregamento excêntrico e apoiadas sobre pilares retangulares", Ph.D. dissertation, Fac. Tecnol., Univ. Brasília, Brasília, 2013. [Online]. Available: https://repositorio.unb.br/handle/10482/13624.
https://repositorio.unb.br/handle/10482/... ].

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Table 9
C5 results for experiments in Trautwein et al. [²⁵25 L. M. Trautwein, R. B. Gomes and G. S. S. A. Melo, "Puncionamento em lajes planas de concreto armado com armadura de cisalhamento interna", vol. 7, no. 1, pp. 38-49, Nov., 2013, http://dx.doi.org/10.5216/reec.v7i1.24333.
http://dx.doi.org/10.5216/reec.v7i1.2433... ].

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Table 10
C5 results for experiments in Albuquerque et al. [²⁶26 E. J. P. Albuquerque, G. S. S. A. Melo, A. M. P. Ramos and V. J. G. Lúcio, "Estudo teórico-experimental da resistência ao punçoamento de lajes fungiformes de betão armado com pilar de canto reentrante sem armadura de punçoamento", In 5ª JPEE, Lisboa, Portugal, 2014. http://dx.doi.org/10.13140/2.1.1473.1849.
http://dx.doi.org/10.13140/2.1.1473.1849... ] and Albuquerque [²⁷27 E. J. P. Albuquerque, "Punção em lajes lisas com armadura de cisalhamento e pilar de canto reentrante", Ph.D. dissertation, Fac. Tecnol., Univ. Brasília, Brasília, 2015. [Online]. Available: https://repositorio.unb.br/handle/10482/18905
https://repositorio.unb.br/handle/10482/... ].

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Table 3
C5 results for experiments in Stamenkovic [¹⁷17 A. Stamenkovic, "Local strength of flat slabs at column heads", Ph.D. thesis, Imperial Coll. of Sci. and Tec., Univ. of London, London, 1969. [Online]. Available: https://spiral.imperial.ac.uk/bitstream/10044/1/16718/2/Stamenkovic-A-1970-PhD-Thesis.pdf
https://spiral.imperial.ac.uk/bitstream/... ] and Stamenkovic and Chapman [¹⁸18 A. Stamenkovic and J. C. Chapman, "Local strength at column heads in flat slabs subjected to a combined vertical and horizontal loading," Proc. Inst. Civ. Eng., vol. 57, pp. 205–232, Jun. 1974, http://dx.doi.org/10.1680/iicep.1974.4054.
http://dx.doi.org/10.1680/iicep.1974.405... ].

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Table 11
C5 results for experiments in Albuquerque [²⁸28 N. G. B. Albuquerque, "Comportamento das ligações de lajes lisas de concreto armado com pilares de borda sujeitas a excentricidades interna e externas", Ph.D. dissertation, Fac. Tecnol., Univ. Brasília, Brasília, 2014. [Online]. Available: https://repositorio.unb.br/handle/10482/17442
https://repositorio.unb.br/handle/10482/... ] and Albuquerque et al. [²⁹29 N. G. B. Albuquerque, G. S. S. A. Melo, and R. L. Vollum, "Punching shear strength of flat slab-edge column connections with outward eccentricity of loading," ACI Struct. J., vol. 113, no. 5, pp. 1117–1129, Sep. 2016, http://dx.doi.org/10.14359/51689156.
http://dx.doi.org/10.14359/51689156... ].

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

Case	Condition	Segment parts	s_P	s_Q	v
1	S < S_A	1	s_I	s_J	-1
2	S = S_A	2	s_I	s_A	-1
2	S = S_A	2	s_A	s_J	1
3	S < S_B	1	s_I	s_J	1
4	S = S_B	2	s_I	s_B	1
4	S = S_B	2	s_B	s_J	-1
5	S > S_B	1	s_I	s_J	-1

	Criteria from the codes
Combinations of criteria from codes	a	b	c	d	e	f	g	h	ψ ≥ 0.95	ψ_mean	ψ _min
Combinations of criteria from codes	ξ	ρ	f_ywd,ef	kd	M*	u ^0*	F_ef	α_θ	ψ ≥ 0.95	ψ_mean	ψ _min
C1	MC90	MC90	MC90	MC90	MC90	MC90	MC90	MC90	64%	1.06	0.12
C2	EC2	EC2	EC2	EC2	EC2	EC2	EC2	EC2	94%	1.29	0.18
C3	NBR 6118	NBR 6118	NBR 6118	NBR 6118	NBR 6118	NBR 6118	NBR 6118	NBR 6118	67%	1.14	0.71
C4	EC2	EC2	EC2	EC2	NBR 6118	EC2	EC2	EC2	99%	1.42	0.92
C5	EC2	EC2	NBR 6118	EC2	NBR 6118	EC2	EC2	EC2	99%	1.42	0.92
C6	NBR 6118	NBR 6118	NBR 6118	EC2	NBR 6118	EC2	EC2	EC2	84%	1.20	0.71
C7	EC2	EC2	NBR 6118	NBR 6118	NBR 6118	EC2	EC2	EC2	97%	1.39	0.89
C8	EC2	EC2	NBR 6118	EC2	NBR 6118	NBR 6118	EC2	EC2	99%	1.42	0.92
C9	EC2	EC2	NBR 6118	EC2	NBR 6118	EC2	MC90	EC2	98%	1.40	0.89
C10	EC2	EC2	NBR 6118	EC2	NBR 6118	EC2	EC2	NBR 6118	98%	1.41	0.82
C5 - Asym.	EC2	EC2	NBR 6118	EC2	NBR 6118	EC2	EC2	EC2	100%	1.66	1.06

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
LC01	CIRCULAR	143	270	270	1.5	48.00	858.4	-	-	858.4	0.79	6	10	70	100	573.0	363.0	436	286	n	1.06
LC02	CIRCULAR	140	360	360	1.55	47.00	955.7	-	-	955.7	0.79	6	10	70	100	573.0	360.0	465	280	n	1.25
LC03	CIRCULAR	142	450	450	1.41	49.00	1076.8	-	-	1076.8	0.79	6	10	70	100	573.0	362.0	493	284	n	1.41
LC05	CIRCULAR	140	360	360	2.05	50.00	1117.5	-	-	1117.5	0.79	6	10	70	100	573.0	360.0	465	280	n	1.32
LC06	CIRCULAR	143	360	360	1.45	49.00	1077.9	-	-	1077.9	0.79	6	10	70	100	573.0	363.0	465	286	n	1.36
LC07	CIRCULAR	144	360	360	1.6	49.00	1110.4	-	-	1110.4	0.79	7	10	55	80	573.0	364.0	442	288	n	1.31
LC08	CIRCULAR	144	360	360	1.62	48.00	1058.9	-	-	1058.9	0.79	6	12	70	100	573.0	364.0	387	288	n	1.06

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
LS01	INTERNAL	145	300	300	1.54	48.00	1021.5	-	-	1021.5	0.79	2	12	70	100	573.0	364.0	177	290	n	1.38
LS02	INTERNAL	143	300	300	1.46	49.00	1127.5	-	-	1127.5	0.79	4	12	70	100	573.0	363.0	278	286	n	1.14
LS03	INTERNAL	145	300	300	1.54	50.00	698.5	-	-189.0	1071.5	0.79	2	12	70	100	573.0	364.0	177	290	n	1.43
LS04	INTERNAL	143	300	300	1.46	49.00	721.7	-	-190.0	1099.9	0.79	4	12	70	100	573.0	363.0	278	286	n	1.12
LS05	INTERNAL	143	300	300	1.58	50.00	779.0	-	-	779.0	-	-	-	-	-	-	-	-	-	1	1.18
LS06	INTERNAL	144	300	300	1.56	50.00	528.3	-	-140.7	807.2	-	-	-	-	-	-	-	-	-	1	1.21
LS07	INTERNAL	143	300	300	1.7	49.00	1196.8	-	-	1196.8	1.23	4	12	70	100	530.0	363.0	280	286	n	1.15
LS08	INTERNAL	144	300	300	1.68	48.00	934.1	-	-190.9	1312.5	1.23	4	12	70	100	530.0	364.0	280	288	n	1.27

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
L1	EDGE	152	300	300	0.75	45.10	293.0	-	-87.8	293.0	-	-	-	-	-	-	-	-	-	1	1.06
L2	EDGE	152	300	300	0.75	45.10	300.0	-	-	300.0	-	-	-	-	-	-	-	-	-	1	1.09
L3	EDGE	152	300	300	0.75	45.10	242.0	-	72.5	493.7	-	-	-	-	-	-	-	-	-	1	1.79
L4	EDGE	152	300	300	0.75	45.10	198.0	-	79.0	472.1	-	-	-	-	-	-	-	-	-	1	1.72

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
L01	REENTRANT	144	300	300	1.4	57.90	300.0	111.4	-111.4	624.7	-	-	-	-	-	-	-	-	-	1	1.23
L02	REENTRANT	144	300	300	1.4	57.90	488.0	120.8	-120.8	840.2	0.5	3	10	70	100	587.0	364.0	231	288	n	1.25
L03	REENTRANT	144	300	300	1.4	57.90	550.0	136.1	-136.1	946.9	0.78	4	10	70	100	562.0	364.0	282	288	n	1.22
L04	REENTRANT	144	300	300	1.4	57.90	347.0	85.9	-85.9	597.4	-	-	-	-	-	-	-	-	-	1	1.18

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
LN01	INTERNAL	143	400	200	1.58	55.10	1084.0	-	-	1084.0	0.5	3	14	70	100	573.0	363.0	201	286	n	1.17
LN02	INTERNAL	143	400	200	1.58	53.80	1144.0	-	-	1144.0	0.5	6	14	70	100	573.0	363.0	334	286	1	1.08
LN03	INTERNAL	143	400	200	1.58	51.20	786.0	-	-	786.0	-	-	-	-	-	-	-	-	-	1	1.18
LN04	INTERNAL	143	400	200	1.58	55.50	966.0	-	-	966.0	0.31	4	14	70	100	651.0	363.0	245	286	1	1.13
LN05	INTERNAL	142	400	200	1.6	54.80	1143.0	-	-	1143.0	1.23	5	14	70	100	602.0	362.0	290	284	n	0.97
LS01	INTERNAL	143	400	200	1.58	53.60	425.0	-	-114.0	673.9	-	-	-	-	-	-	-	-	-	1	0.99
LS02	INTERNAL	144	400	200	1.56	53.90	763.0	-	-218.0	1237.0	0.5	3	14	70	100	573.0	364.0	201	288	n	1.33
LS03	INTERNAL	142	400	200	1.6	54.40	775.0	-	-234.0	1287.9	0.5	6	14	70	100	573.0	362.0	334	284	1	1.23
LS04	INTERNAL	143	400	200	1.58	51.30	712.0	-	-183.0	1111.5	0.31	4	14	70	100	651.0	363.0	245	286	1	1.32
LS05	INTERNAL	142	400	200	1.6	51.00	926.0	-	-272.0	1522.2	1.23	5	14	70	100	602.0	362.0	290	284	n	1.32
LS06	INTERNAL	143	400	200	1.58	53.00	904.0	-	-252.0	1454.1	0.79	6	14	70	100	597.0	363.0	334	286	n	1.27
LW01	INTERNAL	141	200	400	1.62	50.20	446.0	-	-124.0	648.0	-	-	-	-	-	-	-	-	-	1	0.99
LW02	INTERNAL	143	200	400	1.58	52.20	711.0	-	-189.0	1016.0	0.5	3	14	70	100	573.0	363.0	201	286	n	1.11
LW03	INTERNAL	142	200	400	1.6	51.50	733.0	-	-195.0	1049.2	0.5	6	14	70	100	573.0	362.0	319	284	1	1.01
LW04	INTERNAL	142	200	400	1.6	51.50	617.0	-	-131.0	829.4	0.31	4	14	70	100	651.0	362.0	245	284	1	0.99
LW05	INTERNAL	142	200	400	1.6	50.60	815.0	-	-241.0	1205.8	1.23	5	14	70	100	602.0	362.0	290	284	n	1.05

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
L1	INTERNAL	159	200	200	1.2	36.80	1050.0	-	-	1050.0	1.25	11	16	35	60	500.0	375.0	300	318	0	1.10
L4	INTERNAL	164	200	200	1.2	43.40	1038.0	-	-	1038.0	2	11	16	35	60	500.0	379.0	300	328	0	0.92
L9	INTERNAL	154	200	200	1.3	39.40	933.0	-	-	933.0	0.8	11	16	35	60	500.0	371.0	300	308	0	0.95

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
L01	REENTRANT	148	300	300	2.07	48.00	325.0	114.9	-114.9	655.5	-	-	-	-	-	-	-	-	-	1	1.17
L02	REENTRANT	148	300	300	2.07	48.00	513.0	127.7	-127.7	880.3	0.5	3	10	70	100	587.0	367.0	231	296	n	1.19
L03	REENTRANT	145	300	300	2.17	48.00	575.0	141.9	-141.9	987.3	0.78	4	10	70	100	560.0	364.0	282	290	n	1.19
L04	REENTRANT	147	300	300	2.1	48.00	372.0	91.8	-91.8	637.0	-	-	-	-	-	-	-	-	-	1	1.15
L05	REENTRANT	143	300	300	0.91	44.00	250.0	89.1	-89.1	510.7	-	-	-	-	-	-	-	-	-	1	1.29
L06	REENTRANT	145	300	300	0.88	44.00	282.0	68.4	-68.4	480.8	-	-	-	-	-	-	-	-	-	1	1.20
L07	REENTRANT	141	300	300	1.35	44.00	358.0	88.3	-88.3	618.3	-	-	-	-	-	-	-	-	-	1	1.40
L08	REENTRANT	146	300	300	1.27	44.00	345.0	84.7	-84.7	590.2	-	-	-	-	-	-	-	-	-	1	1.29
L09	REENTRANT	148	300	300	2.08	43.00	550.0	135.7	-135.7	940.4	0.78	5	13	60	90	528.0	367.0	250	296	n	1.07
L10	REENTRANT	148	300	300	2.08	43.00	500.0	176.1	-176.1	1006.5	0.78	5	10	70	90	528.0	367.0	334	296	n	1.28
L11	REENTRANT	147	300	300	2.11	43.00	640.0	120.8	-120.8	988.7	0.78	5	10	70	90	528.0	366.0	334	294	n	1.27
L12	REENTRANT	145	300	300	1.28	43.00	345.0	65.4	-65.4	535.0	-	-	-	-	-	-	-	-	-	1	1.19

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
V_I_1	INTERNAL	56	127	127	1.17	40.14	119.7	-	-	119.7	-	-	-	-	-	-	-	-	-	1	1.36
V_I1_1	INTERNAL	56	127	127	1.17	38.68	104.5	-	-	104.5	-	-	-	-	-	-	-	-	-	1	1.20
V_I2_1	INTERNAL	56	127	127	2.34	56.26	129.9	-	-	129.9	-	-	-	-	-	-	-	-	-	1	1.10
V_I_2	INTERNAL	56	127	127	1.17	27.54	117.4	-	-	117.4	-	-	-	-	-	-	-	-	-	1	1.51
V_Ir_1	INTERNAL	56	152	76	1.17	26.72	108.5	-	-	108.5	-	-	-	-	-	-	-	-	-	1	1.48
V_E_1	EDGE	56	127	127	1.17	30.47	74.7	-	-	74.7	-	-	-	-	-	-	-	-	-	1	1.86
V_C_1	CORNER	56	127	127	1.17	34.23	27.1	-	-	27.1	-	-	-	-	-	-	-	-	-	1	1.30
C_I_1	INTERNAL	56	127	127	1.17	38.27	84.5	-	-7.3	120.6	-	-	-	-	-	-	-	-	-	1	1.39
C_I_2	INTERNAL	56	127	127	1.17	31.53	62.3	-	-10.5	114.0	-	-	-	-	-	-	-	-	-	1	1.41
C_I_3	INTERNAL	56	127	127	1.17	27.13	33.8	-	-13.6	101.1	-	-	-	-	-	-	-	-	-	1	1.31
C_I_4	INTERNAL	56	127	127	1.17	26.67	20.9	-	-16.7	103.1	-	-	-	-	-	-	-	-	-	1	1.34
C_Ir_1	INTERNAL	56	152	76	1.17	24.03	85.7	-	-7.3	127.0	-	-	-	-	-	-	-	-	-	1	1.79
C_Ir_2	INTERNAL	56	152	76	1.17	31.06	67.3	-	-10.9	128.6	-	-	-	-	-	-	-	-	-	1	1.66
C_Ir_3	INTERNAL	56	152	76	1.17	30.36	39.9	-	-15.7	128.6	-	-	-	-	-	-	-	-	-	1	1.68
C_Ir_4	INTERNAL	56	152	76	1.17	28.25	21.6	-	-16.8	116.5	-	-	-	-	-	-	-	-	-	1	1.56
Ct_E_1	EDGE	56	127	127	1.17	29.60	45.8	4.9	-	59.6	-	-	-	-	-	-	-	-	-	1	1.50
Ct_E_2	EDGE	56	127	127	1.17	30.18	34.9	5.7	-	51.1	-	-	-	-	-	-	-	-	-	1	1.28
Ct_E_3	EDGE	56	127	127	1.17	29.65	23.5	9.4	-	50.1	-	-	-	-	-	-	-	-	-	1	1.26
Ct_E_4	EDGE	56	127	127	1.17	31.06	12.9	10.1	-	41.6	-	-	-	-	-	-	-	-	-	1	1.03
Cn_E_1	EDGE	56	127	127	1.17	32.70	73.2	-	-5.6	73.2	-	-	-	-	-	-	-	-	-	1	1.78
Cn_E_2	EDGE	56	127	127	1.17	27.54	54.7	-	-9.2	76.4	-	-	-	-	-	-	-	-	-	1	1.97
Cn_E_3	EDGE	56	127	127	1.17	28.89	24.9	-	-10.1	85.5	-	-	-	-	-	-	-	-	-	1	2.17
Cn_E_4	EDGE	56	127	127	1.17	29.19	10.9	-	-8.8	75.7	-	-	-	-	-	-	-	-	-	1	1.91
C_C_1	CORNER	56	127	127	1.17	32.35	24.9	-	-6.2	56.7	-	-	-	-	-	-	-	-	-	1	2.78
C_C_2	CORNER	56	127	127	1.17	30.06	15.9	-	-6.4	60.0	-	-	-	-	-	-	-	-	-	1	3.01
C_C_3	CORNER	56	127	127	1.17	27.43	8.0	-	-6.2	59.6	-	-	-	-	-	-	-	-	-	1	3.08
C_C_4	CORNER	56	127	127	1.17	32.53	3.6	-	-5.6	55.1	-	-	-	-	-	-	-	-	-	1	2.69

Exp.	Case	d	c_x	c_y	ρ	f_ck	F	M_x	M_y	$F_{e f}^{1}$	$A_{s w}^{ϕ}$	n_p	n_r	s ₀	s_r	f_ywk	f_ywk,ef	s_avg	s_max	Crit. per.	ψ
Exp.	Case	(mm)	(mm)	(mm)	(%)	(MPa)	(kN)	(kNm)	(kNm)	(kN)	(cm²)	n_p	n_r	(mm)	(mm)	(MPa)	(MPa)	(mm)	(mm)	Crit. per.	ψ
L1	EDGE	147	300	300	1	46.80	308.0	-	-92.0	308.0	-	-	-	-	-	-	-	-	-	1	1.06
L2	EDGE	146	300	300	1.25	44.70	315.0	-	-	315.0	-	-	-	-	-	-	-	-	-	1	1.03
L3	EDGE	146	300	300	1.25	45.10	256.0	-	77.0	527.2	-	-	-	-	-	-	-	-	-	1	1.72
L4	EDGE	146	300	300	1.25	46.00	210.0	-	84.0	505.8	-	-	-	-	-	-	-	-	-	1	1.64
L5	EDGE	146	300	300	1.25	51.40	374.0	-	37.0	504.3	-	-	-	-	-	-	-	-	-	1	1.58
L6	EDGE	146	300	300	1.25	52.10	330.0	-	66.0	562.4	-	-	-	-	-	-	-	-	-	1	1.75
L7	EDGE	146	300	300	1.52	50.00	288.0	-	115.0	693.0	-	-	-	-	-	-	-	-	-	1	2.05
L8	EDGE	146	300	300	1.4	50.50	320.0	-	128.0	770.8	-	-	-	-	-	-	-	-	-	1	2.34
L9	EDGE	146	300	300	1.25	57.60	489.0	-	-	489.0	0.5	4	7	70	100	580.0	365.0	287	292	n	0.96
L10	EDGE	146	300	300	1.52	59.30	445.0	-	89.0	758.4	0.5	4	7	70	100	580.0	365.0	287	292	n	1.39
L11	EDGE	146	300	300	1.52	43.10	304.0	-	110.0	691.4	-	-	-	-	-	-	-	-	-	1	2.15
L12	EDGE	146	300	300	1.52	43.60	347.0	-	55.0	540.7	-	-	-	-	-	-	-	-	-	1	1.68
L13	EDGE	146	300	300	1.52	44.10	357.0	-	125.0	797.2	-	-	-	-	-	-	-	-	-	1	2.46

\begin{matrix} \cos α = \frac{M_{x}}{\sqrt{M_{x}^{2} + M_{y}^{2}}} & ; & \sin α = \frac{M_{y}}{\sqrt{M_{x}^{2} + M_{y}^{2}}} \end{matrix}

M_{\bar{x}} = \int_{U} v \bar{y} d s = \int_{U} v (- x \sin α + y \cos α) d s = M_{x} \cos α + M_{y} \sin α = \sqrt{M_{x}^{2} + M_{y}^{2}}

\begin{matrix} ζ_{G I} = \frac{(s_{J} - s_{G})}{(s_{J} - s_{I})} & ; & ζ_{G J} = \frac{(s_{G} - s_{I})}{(s_{J} - s_{I})} \end{matrix}

\begin{matrix} {\bar{x}}_{H} = ζ_{H I} {\bar{x}}_{I} + ζ_{H J} {\bar{x}}_{J} & ; & {\bar{y}}_{H} = ζ_{H I} {\bar{y}}_{I} + ζ_{H J} {\bar{y}}_{J} \end{matrix}

W_{p \bar{x}}^{P Q} = \int_{α_{P}}^{α_{Q}} 1 \bar{y} d s = \int_{α_{P}}^{α_{Q}} 1 ({\bar{y}}_{C} + r \sin α) r d α = Δ s {\bar{y}}_{C} + r^{2} (\cos α_{Q} - \cos α_{P})

W_{p \bar{y}}^{P Q} = - \int_{α_{P}}^{α_{Q}} 1 \bar{x} d s = - \int_{α_{P}}^{α_{Q}} 1 ({\bar{x}}_{C} + r \cos α) r d α = - Δ s {\bar{x}}_{C} - r^{2} (\sin α_{Q} - \sin α_{P})

[1] Corresponding author: Mauro Schulz. E-mail: mschulz@id.uff.br

Brasil

Brasil

Punching shear design with control perimeters subjected to asymmetrical bending

Verificação à punção com perímetros de controle submetidos à flexão assimétrica

Abstract