Reliability of codes provisions for punching shear design

This paper presents a reliability analysis of the punching shear of flat slabs without shear reinforcement. The evaluation is performed for the codes ACI 318, Eurocode 2, Model Code 2010, and ABNT NBR 6118:2014. Six models were used for predicting the punching strength, the design models, the Critical Shear Crack Theory (CSCT), and a Nonlinear Finite Element Analysis (NLFEA) model. Reliability analysis was performed to evaluate the safety level of code provisions. Design code provisions were evaluated in terms of sufficient reliability criteria based on target reliability β=3.8. This target reliability was based in the Model Code 2010 criteria. This criterion represents an acceptable value level of safety in design of structures. The results showed that the reliability indexes β presented satisfactory for all slabs designed by the Model Code 2010. However, this paper also shows some of the tested slabs presented reliability index below 3.0, being the ACI 318 the code with the lowest reliability index.

Keywords:
structural reliability; punching shea strength; resistance error model; flat slabs; design codes

Ref.	Slabs	$V_{e x p}$ (kN)	$V_{e x p} / V_{R}$	EC2	ACI 318	MC 2010	CSCT	NLFEA
Ref.	Slabs	$V_{e x p}$ (kN)	NBR	EC2	ACI 318	MC 2010	CSCT	NLFEA
[¹¹11 B. R. C. Elstner and E. Hognestadt, “Shearing strength of reinforced concrete slabs,” ACI J. Proc., vol. 53, no. 7, pp. 29–58, 1956, http://dx.doi.org/10.14359/11501. http://dx.doi.org/10.14359/11501... ]	A-1b	365	0.96	1.10	1.24	1.18	1.03	1.02
	A-1c	356	0.89	1.03	1.13	1.10	0.96	0.98
	A-1d	351	0.81	0.94	0.99	1.00	0.87	0.90
	A-1e	356	1.01	1.16	1.35	1.24	1.08	1.00
	A-2b	400	0.94	1.17	1.62	1.21	1.06	1.02
	A-2c	467	0.88	1.10	1.37	1.09	0.95	0.87
	A-7b	512	1.06	1.33	1.73	1.34	1.17	1.11
	A-3b	445	0.87	1.24	1.67	1.17	1.03	0.94
	A-3c	534	0.99	1.41	1.85	1.31	1.15	1.03
	A-3d	547	0.93	1.33	1.66	1.19	1.05	0.93
	A-4	400	0.89	1.03	1.05	1.07	0.93	0.96
	A-5	534	0.95	1.19	1.42	1.15	1.00	0.86
	B-9	505	0.97	1.13	1.36	1.18	0.97	0.95
	B-14	578	0.93	1.24	1.45	1.13	0.95	1.03
[¹²12 P. Regan, Behaviour of Reinforced Concrete Flat Slabs. London, United Kingdom: Constr. Ind. Res. Inf. Assoc., 1981.]	I/2	176	0.91	1.18	1.28	1.40	1.22	1.09
	II/1	825	1.00	1.00	1.16	1.26	1.10	0.92
	II/2	390	1.04	1.17	1.37	1.32	1.15	1.03
	II/3	365	0.96	1.08	1.27	1.33	1.16	0.95
	II/4	117	1.01	1.40	1.65	1.34	1.18	1.16
	II/5	105	0.90	1.25	1.46	1.28	1.12	1.03
	II/6	105	0.89	1.23	1.42	1.33	1.16	1.01
[¹³13 H. Marzouk and A. Hussein, “Experimental investigation on the behavior of high-strength concrete slabs,” ACI Struct. J., vol. 88, no. 6, pp. 701–713, Nov./Dec. 1991.]	HS2	249	0.91	1.11	0.96	1.22	1.06	1.02
	HS3	356	1.09	1.33	1.38	1.39	1.21	0.90
	HS4	418	1.16	1.46	1.79	1.48	1.30	1.05
	HS7	356	1.17	1.43	1.33	1.53	1.33	1.03
	HS8	436	1.02	1.17	1.22	1.25	1.09	0.98
	HS9	543	1.08	1.24	1.46	1.31	1.15	0.98
	HS10	645	1.12	1.30	1.67	1.36	1.19	1.10
	HS12	258	1.15	1.54	1.45	1.60	1.39	1.13
	HS13	267	1.12	1.50	1.58	1.52	1.33	0.89
	HS14	498	1.30	1.59	1.47	1.60	1.40	0.97
	HS15	560	1.27	1.55	1.33	1.54	1.34	1.11
	NS1	320	1.15	1.41	1.59	1.47	1.28	0.98
[¹⁴14 A. Tomaszewicz, High-Strength Concrete. SP2 – Plates and Shells. Report 2.3 Punching Shear Capacity of Reinforced Concrete Slabs. Trondheim, Norway: SINTEF, 1993. [Report no. STF70 A93082].]	ND65-2-1	1200	1.09	1.09	1.53	1.38	1.21	1.00
	ND95-1-1	2250	1.15	1.15	1.45	1.40	1.22	0.99
	ND95-1-3	2400	1.01	1.09	1.49	1.23	1.07	0.87
	ND95-2-1	1100	0.92	0.92	1.25	1.17	1.03	0.86
	ND95-2-1D	1300	1.10	1.10	1.50	1.39	1.22	1.01
	ND95-2-3	1450	1.05	1.15	1.64	1.34	1.18	0.97
	ND95-2-3D	1250	0.94	1.03	1.49	1.21	1.06	0.86
	ND95-2-3D+	1450	1.02	1.11	1.57	1.30	1.14	0.94
	ND95-3-1	330	1.03	1.29	1.62	1.30	1.14	1.10
	ND115-1-1	2450	1.14	1.14	1.36	1.39	1.21	1.01
	ND115-2-1	1400	1.06	1.06	1.38	1.35	1.18	0.97
	ND115-2-3	1550	1.06	1.15	1.60	1.34	1.17	0.97
[¹⁵15 S. Guandalini, O. L. Burdet, and A. Muttoni, “Punching tests of slabs with low reinforcement ratios,” ACI Struct. J., vol. 106, pp. 87–95, 2009, http://dx.doi.org/10.14359/56287. http://dx.doi.org/10.14359/56287... ]	PG-1	1023	1.08	1.08	1.48	1.41	1.23	1.11
	PG-11	763	0.97	0.97	1.03	1.31	1.14	0.98
	PG-6	238	0.87	1.07	1.40	1.19	1.04	0.84
	PG-7	241	1.05	1.27	1.33	1.43	1.24	1.11
[¹⁶16 S. Teng, H. K. Cheong, K. L. Kuang, and J. Z. Geng, “Punching shear strength of slabs with openings and supported on rectangular columns,” ACI Struct. J., vol. 101, no. 5, pp. 678–687, 2004.]	OC11	423	0.87	0.99	1.30	1.19	1.04	1.12
[¹⁷17 R. B. Gomes and P. E. Regan, “Punching resistance of RC flat slabs with shear reinforcement,” J. Struct. Eng., vol. 125, pp. 684–692, Jun. 1999.]	1	560	0.84	0.89	1.16	1.14	1.00	0.91
	1A	587	0.87	0.93	1.20	1.19	1.04	0.95
[¹⁸18 A. E. G. Coelho, “Puncionamento em lajes cogumelo de concreto armado com resistência de 30 MPa e armadura de cisalhamento vertical e inclinada,” Master thesis, Univ. Brasília, Brasília, Brazil, 1999. [Online]. Available: https://repositorio.unb.br/handle/10482/37408 https://repositorio.unb.br/handle/10482/... ]	L1	316	1.15	1.35	1.77	1.48	1.30	1.08
[¹⁹19 J. L. P. Melges, “Análise experimental da punção em lajes de concreto armado e protendido,” Doctoral dissertation, Univ. São Paulo, São Carlos, Brazil, 2001, http://dx.doi.org/10.11606/T.18.2001.tde-07062006-152744. http://dx.doi.org/10.11606/T.18.2001.tde... ]	M1	441	0.99	1.09	1.48	1.28	1.12	1.15
[²⁰20 T. H. Musse, “Punção em lajes cogumelo: fibras de aço e armadura de cisalhamento,” Ph.D. dissertation, Escola Eng. Civil, Univ. Fed. Goiás, Goiânia, Brazil, 2004.]	L1	309	1.11	1.38	1.63	1.47	1.29	1.00
[²¹21 B. Adetifa and M. A. Polak, “Retrofit of slab column interior connections using shear bolts,” ACI Struct. J., vol. 102, pp. 268–274, 2005, http://dx.doi.org/10.14359/14278. http://dx.doi.org/10.14359/14278... ]	SB1	253	0.98	1.23	1.35	1.32	1.06	0.99
[²²22 J. Sagaseta, L. Tassinari, M. F. Ruiz, and A. Muttoni, “Punching of flat slabs supported on rectangular columns,” Eng. Struct., vol. 77, pp. 17–33, Oct. 2014, http://dx.doi.org/10.1016/j.engstruct.2014.07.007. https://doi.org/10.1016/j.engstruct.2014... ]	PT31	1433	1.09	1.09	1.32	1.42	1.27	0.97
[²³23 M. G. Marques, “Punção em lajes lisas de concreto armado com aberturas adjacentes ao pilar e armadura de cisalhamento,” Doctoral dissertation, Facul. Eng. Civil, Arquit. Urban., Univ. Estadual Campinas, Campinas, Brazil, 2018. [Online]. Available: https://hdl.handle.net/20.500.12733/1635317 https://hdl.handle.net/20.500.12733/1635... ]	LR	232.3	0.84	1.04	1.28	1.11	0.97	0.89
[²⁴24 E. A. P. Liberati, “Análise de confiabilidade de lajes lisas de concreto armado submetidas à punção,” Doctoral dissertation, Facul. Eng. Civil, Arquit. Urban., Univ. Estadual Campinas, Campinas, Brazil, 2019. [Online]. Available: https://hdl.handle.net/20.500.12733/1635891 https://hdl.handle.net/20.500.12733/1635... ]	LR_A	249.9	0.89	1.12	1.37	1.24	1.08	1.01
	LR_B	216.4	0.78	0.98	1.20	1.09	0.95	0.85
	LR_C	259.2	0.92	1.15	1.40	1.28	1.11	0.98
[²⁵25 A. W. Lima, “Estudo experimental e analítico de punção em lajes lisas de concreto armado utilizando polímeros reforçados com fibra de carbono (PRFC) como armadura de cisalhamento,” Master thesis, Facul. Technol., Univ. Brasília, Brasília, Brazil, 2012. [Online]. Available: https://repositorio.unb.br/handle/10482/11167 https://repositorio.unb.br/handle/10482/... ]	L0-01	571	0.84	0.92	1.02	1.00	0.87	1.12
[²⁶26 M. P. Ferreira, “Punção em lajes lisas de concreto armado com armaduras de cisalhamento e momentos desbalanceados,” Doctoral dissertation, Univ. Brasília, Brasília, Brazil, 2010. [Online]. Available: https://repositorio.unb.br/handle/10482/8965 https://repositorio.unb.br/handle/10482/... ]	LS-05	779	1.07	1.17	1.30	1.34	1.17	1.05
[²⁷27 T. Clément, A. P. Ramos, M. Fernández Ruiz, and A. Muttoni, “Influence of prestressing on the punching strength of post-tensioned slabs,” Eng. Struct., vol. 72, pp. 56–69, Aug. 2014, http://dx.doi.org/10.1016/j.engstruct.2014.04.034. http://dx.doi.org/10.1016/j.engstruct.20... ]	PG19	860	0.97	0.97	0.99	1.30	1.13	0.97
	PG20	1094	0.98	0.98	1.23	1.24	1.08	0.96
		Mean	1.00	1.17	1.40	1.29	1.13	0.99
		CoV	0.11	0.14	0.15	0.11	0.11	0.08

Model	$β_{0}$	$β_{1}$	$β_{2}$	$β_{3}$	$β_{4}$	$β_{5}$	$β_{6}$	$ε ~ N (μ, σ)$
Model	$β_{0}$	$β_{1}$	$β_{2}$	$β_{3}$	$β_{4}$	$β_{5}$	$β_{6}$	$μ$	$σ$
ABNT NBR 6118	-0.1086	0.0021	-	-	-	-	-	0.0	0.0976
EUROCODE 2	0.3255	-	-	-	-0.0013	0.0045	-	0.0	0.1456
ACI 318	0.3162	-	0.1444	-0.0012	-	-	-	0.0	0.1394
CSCT	0.0350	0.0010	-	-0.0003	-	-	0.0002	0.0	0.1111
MC 2010	0.1892	0.0010	-	-0.0004	-	-	0.0002	0.0	0.1256
NLFEA	0.0658	-	-0.0214	-	-0.0003	-	-	0.0	0.0777

Random variable	Distribution	Mean	Cov.	Reference
Concrete compressive strength $(f_{c})$	Normal	$1.22 f_{c k}$ (MPa)	0.15	[²⁸28 W. C. Santiago, “Calibração baseada em confiabilidade dos coeficientes parciais de segurança das principais normas brasileiras de projeto estrutural,” Doctoral dissertation, Escola Eng. São Carlos, Univ. São Paulo, São Carlos, Brazil, 2019, http://dx.doi.org/10.11606/T.18.2019.tde-03042019-092619. http://dx.doi.org/10.11606/T.18.2019.tde... ]
Clear cover of reinforcement $(d')$	Normal	$d'$ (mm)	0.125	[²⁹29 F. R. Stucchi, S. H. C. Santos, and R. M. Franco, “Evaluation of the safety criteria of Brazilian Standard NBR 6118 for slender members based on reliability analyses,” in 11th Int. Conf. Appl. Statist. Probability Civil Eng., M. Faber, J. Koehler, and K. Nishijima, Eds. Aug. 2011, pp. 2761–2767.]
Slab thickness $(h)$	Normal	$h$ (mm)	0.4/ $h$ +0.006	[³⁰30 Joint Committee on Structural Safety. “Probabilistic model code – part 2: load models.” JCSS. https://www.jcss-lc.org/jcss-probabilistic-model-code/ (accessed Jul. 3, 2023). https://www.jcss-lc.org/jcss-probabilist... ]
Steel yield strength $(f y)$	Normal	$1.12 f_{y k}$ (MPa)	0.05	[³⁰30 Joint Committee on Structural Safety. “Probabilistic model code – part 2: load models.” JCSS. https://www.jcss-lc.org/jcss-probabilistic-model-code/ (accessed Jul. 3, 2023). https://www.jcss-lc.org/jcss-probabilist... ]
Residual of the resistance model error $(ε)$	Normal		Following Table 2
Dead load $(D)$	Normal	$1.06 D_{n}$	0.12	[³¹31 L. G. L. Costa, W. C. Santiago, and A. T. Beck, “Probabilistic models for live loads in buildings: critical review, comparison to Brazilian design standards and calibration of partial safety factors,” Rev. IBRACON Estrut. Mater., vol. 16, no. 2, pp. e16204, Sep. 2023. Accessed: Jul. 3, 2023. [Online]. Available: https://www.scielo.br/j/riem/a/gT97H7GRzn4fRKV4MHh9PZL/?lang=en https://www.scielo.br/j/riem/a/gT97H7GRz... ]
Live load $(L)$	Gumbel	$0.92 L_{n}$	0.25	[³¹31 L. G. L. Costa, W. C. Santiago, and A. T. Beck, “Probabilistic models for live loads in buildings: critical review, comparison to Brazilian design standards and calibration of partial safety factors,” Rev. IBRACON Estrut. Mater., vol. 16, no. 2, pp. e16204, Sep. 2023. Accessed: Jul. 3, 2023. [Online]. Available: https://www.scielo.br/j/riem/a/gT97H7GRzn4fRKV4MHh9PZL/?lang=en https://www.scielo.br/j/riem/a/gT97H7GRz... ]
Load model error $(E_{m, S})$	Lognormal	1.00	0.10	[³⁰30 Joint Committee on Structural Safety. “Probabilistic model code – part 2: load models.” JCSS. https://www.jcss-lc.org/jcss-probabilistic-model-code/ (accessed Jul. 3, 2023). https://www.jcss-lc.org/jcss-probabilist... ]

Design code	Load combinations
ABNT NBR 6118:2014	$F_{s d} = 1.4 D_{n} + 1.4 L_{n}$
ACI 318	$F_{s d} \geq \{\begin{matrix} 1.4 D_{n} \\ 1.2 D_{n} + 1.6 L_{n} \end{matrix}$
EC 2	$F_{s d} \geq \{\begin{matrix} 1.15 D_{n} + 1.50 L_{n} \\ 1.35 D_{n} + 1.05 L_{n} \end{matrix}$
MC 2010	$F_{s d} = 1.15 D_{n} + 1.50 L_{n}$

Design code	βmin	βmax	βrange
ACI 318	2.42	4.19	1.77
NBR 6118:2014	3.23	3.50	0.27
MC 2010	4.23	5.58	1.35
EC2	3.51	3.85	0.34

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[1] Corresponding author: Gustavo Ribeiro da Silva. E-mail: gustavoribeirods@gmail.com