Reliability analysis of slender columns using the general method with linear creep theory

Ribeiro, Kleyser; Loriggio, Daniel Domingues; Real, Mauro de Vasconcellos

doi:10.1590/S1983-41952022000300008

Kleyser Ribeiro

Universidade Federal de Santa Catarina – UFSC, Florianópolis, SC, Brasil

https://orcid.org/0000-0002-2310-1563

Daniel Domingues Loriggio

Universidade Federal de Santa Catarina – UFSC, Florianópolis, SC, Brasil

https://orcid.org/0000-0002-1032-4805

Mauro de Vasconcellos Real

Universidade Federal do Rio Grande – FURG, Rio Grande, RS, Brasil

https://orcid.org/0000-0003-4916-9133

Corresponding author: Kleyser Ribeiro. E-mail: kleyser.ribeiro@gmail.com

Conflict of interest: Nothing to declare.

Author contributions: KR: conceptualization, methodology, computational programming, numerical tests, results analysis, writing; DDL: conceptualization, methodology, results analysis, supervision; MVR: conceptualization, methodology, results analysis, supervision.

Editors: Yuri Villagrán Zaccardi, Guilherme Aris Parsekian.

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Figure 1
Flowchart of the moment-curvature relationship.

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Figure 2
Moment-curvature diagram according to the Brazilian Standard NBR 6118 [¹1 Associação Brasileira de Normas Técnicas, Projeto de Estruturas de Concreto – Procedimento, NBR 6118:2014, 2014.].

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Figure 3
Cantilever column for application of the finite difference method.

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Figure 4
Flowchart of the finite difference method.

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Figure 5
Simplified stress-strain diagram of concrete with creep.

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Figure 6
Simplified stress-strain diagram for longitudinal reinforcement.

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Figure 7
Constitutive model for non-linear analysis with creep.

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Figure 8
Constitutive model for tension in concrete with tension-stiffening effect.

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Figure 9
Rectangular block corresponding to the effective tension area.

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Figure 10
Reliability analysis process by Monte Carlo and FORM.

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Figure 11
Flowchart of the FORM algorithm.

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Figure 12
Distribution of database variables.

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Figure 13
Cross-sections of the columns considered in the research.

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Figure 14
Comparative analysis of results.

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Figure 15
Analysis of load effect and resistance by probability density function (PDF).

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Table 1
Random variables.

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Table 2
Main characteristics of the columns considered in the model validation.

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Table 3
Statistical synthesis of results of the validation process.

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Table 4
Reliability index of the tested columns.

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Table 5
Analysis of load effect, resistance, and safety margin.

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Random variable	Distribution	Mean (μ)	Standard deviation (σ)
Compressive strength of concrete (f_c)	Normal	$μ_{f_{c}} = \frac{f_{ck}}{1 - 1 .645 V_{f_{c}}}$ where $V_{f_{c}} = 0 .10$	$σ_{f_{c}} {= μ}_{f_{c}} V_{f_{c}}$ where $V_{f_{c}} = 0 .10$
Yield strength of reinforcement (f_y)	Normal	$μ_{f_{y}} = 1 {.09f}_{yk}$	$σ_{f_{y}} = 0 {.05μ}_{f_{y}}$
Width of the cross-section (b)	Normal	$μ_{b} = b$ (nominal design value)	$σ_{b} = 0 .5$ cm
Overall depth of the cross-section (h)	Normal	$μ_{h} = h$ (nominal design value)	$σ_{h} = 0 .5$ cm
Effective depth of the cross-section (d), taking d < h	Normal	$μ_{d} = d$ (nominal design value)	$σ_{d} = 0 .5$ cm
Permanent load (F_g)	Normal	$μ_{F_{g}} = 1 {.05F}_{gk}$	$σ_{F_{g}} = 0 {.10μ}_{F_{g}}$
Variable load (F_q)	Gumbel max.	$μ_{F_{q}} {= F}_{qk}$ (characteristic value)	$σ_{F_{q}} = 0 {.25μ}_{F_{q}}$

Reference	n	Cross-section (cm x cm)	f_c (MPa)	ρ (%)	λ	e₁/h
Short-term loading
Claeson and Gylltoft [¹⁵15 C. Claeson and K. Gylltoft, "Slender high-strength concrete columns subjected to eccentric loading," J. Struct. Eng., vol. 124, pp. 233–240, Mar 1998, http://dx.doi.org/10.1061/(ASCE)0733-9445(1998)124:3(233). http://dx.doi.org/10.1061/(ASCE)0733-944... ]	12	12×12 or 20×20	33.0 to 93.0	1.99 to 3.11	52 to 69	0.10 to 0.17
Dantas [¹⁶16 A. B. Dantas, “Estudo de pilares de concreto armado submetidos à flexão composta reta,” M.S. thesis, PECC, UnB, Brasília, 2006.]	5	25×12	33.9 to 37.6	1.57	87	0.12 to 0.50
Enciso [¹⁷17 R. O. Enciso, “Comportamento de pilares esbeltos de concreto de alta resistência sujeitos à flexão composta reta,” M.S. thesis, PPGEC, PUC-Rio, Rio de Janeiro, 2010.]	4	25×15	46.9 to 53.6	1.26 to 4.29	69	0.13
Goyal and Jackson [¹⁸18 B. B. Goyal and N. Jackson, "Slender concrete columns under sustained load," J. Struct. Div., vol. 97, no. 11, pp. 2729–2750, Nov 1971, http://dx.doi.org/10.1061/JSDEAG.0003050. http://dx.doi.org/10.1061/JSDEAG.0003050... ]	26	7.62×7.62	19.9 to 23.6	1.72 to 2.45	55 to 125	0.17 to 0.50
Kim and Lee [¹⁹19 J.-K. Kim and S.-S. Lee, "The behavior of reinforced concrete columns subjected to axial force and biaxial bending," Eng. Struct., vol. 22, no. 11, pp. 1518–1528, Nov 2000, http://dx.doi.org/10.1016/S0141-0296(99)00090-5. http://dx.doi.org/10.1016/S0141-0296(99)... ]	4	10×10 or 20×10	27.0	2.13 to 2.84	42	0.40
Kim and Yang [²⁰20 J.-K. Kim and J.-K. Yang, "Buckling behaviour of slender high-strength concrete columns," Eng. Struct., vol. 17, no. 1, pp. 39–51, 1995, http://dx.doi.org/10.1016/0141-0296(95)91039-4. http://dx.doi.org/10.1016/0141-0296(95)9... ]	18	8×8	25.5 to 86.2	1.98 to 3.95	62 to 104	0.30
Melo [²¹21 C. E. L. Melo, “Análise experimental e numérica de pilares birrotulados de concreto armado submetidos a flexo-compressão normal,” Ph.D. dissertation, PECC, UnB, Brasília, 2009.]	17	25×12	37.2 to 45.8	1.57	58 to 87	0.05 to 0.50
Sustained loading
Goyal and Jackson [¹⁸18 B. B. Goyal and N. Jackson, "Slender concrete columns under sustained load," J. Struct. Div., vol. 97, no. 11, pp. 2729–2750, Nov 1971, http://dx.doi.org/10.1061/JSDEAG.0003050. http://dx.doi.org/10.1061/JSDEAG.0003050... ]	20	7.62×7.62	19.9 to 23.6	1.72 to 2.45	55 to 125	0.17 to 0.50
Kordina [²²22 K. Kordina, Langzeitversuche an Stahlbetonstützen. Braunschweig: Inst. Baustoffkunde und Stahlbetonbau, Tech. Univ. Braunschweig, June 1972.]	10	26.4 ≤ b ≤ 27.2 17.2 ≤ h ≤ 17.6	20.9 to 27.1	0.98 to 3.19	101 to 104	0.20 to 0.50
Ramu et al. [²³23 P. Ramu, M. Grenacher, M. Baumann, and B. Thürlimann, Versuche an Gelenkig Gelagerten Stahlbetonstützen unter Dauerlast. Zürich: Institut für Baustatik, Eidgenössische Technische Hochschule Zürich, 1969, https://doi.org/10.3929/ethz-a-004170564. https://doi.org/10.3929/ethz-a-004170564... ]	8	25×15	21.5 to 37.2	1.66 to 4.21	100	0.03 to 0.25

Ratio	Minimum	Maximum	Range	Mean	Standard deviation	V_model
F_exp / F_mod	0.75	1.31	0.56	0.99	0.10	0.08

Cross-section	e₁/h	ρ (%)	f_ck = 30 MPa			f_ck = 60 MPa
			Slenderness ratio (λ)			Slenderness ratio (λ)
			100	120	140	100	120	140
S1	0.12	0.75	4.15	4.15	4.26	3.87	3.93	3.99
		1.50	3.77	3.84	3.82	3.65	3.67	3.71
		2.25	3.53	3.53	3.56	3.48	3.47	3.50
		3.00	3.47	3.32	3.36	3.37	3.33	3.35
	0.18	0.75	4.02	4.14	4.27	3.99	4.16	4.31
		1.50	3.58	3.57	3.63	3.52	3.59	3.70
		2.25	3.47	3.38	3.29	3.36	3.31	3.32
		3.00	3.51	3.30	3.21	3.35	3.23	3.17
	0.24	0.75	3.88	3.98	4.02	4.01	4.11	4.36
		1.50	3.60	3.49	3.41	3.50	3.51	3.53
		2.25	3.49	3.38	3.31	3.45	3.35	3.29
		3.00	3.54	3.29	3.26	3.29	3.30	3.19
S2	0.12	0.75	4.27	4.33	4.41	3.97	4.04	4.17
		1.50	3.96	4.04	4.04	3.81	3.86	3.88
		2.25	3.71	3.73	3.77	3.65	3.64	3.70
		3.00	3.53	3.52	3.56	3.47	3.47	3.51
	0.18	0.75	4.26	4.38	4.50	4.16	4.35	4.59
		1.50	3.75	3.77	3.81	3.73	3.80	3.93
		2.25	3.56	3.52	3.50	3.48	3.48	3.52
		3.00	3.52	3.38	3.39	3.39	3.33	3.33
	0.24	0.75	4.08	4.16	4.34	4.21	4.38	4.65
		1.50	3.75	3.61	3.67	3.62	3.74	3.71
		2.25	3.57	3.54	3.47	3.55	3.45	3.48
		3.00	3.53	3.43	3.39	3.47	3.42	3.32
S3	0.12	0.75	4.30	4.40	4.45	3.99	4.01	4.20
		1.50	4.00	4.07	4.09	3.80	3.88	3.91
		2.25	3.74	3.79	3.83	3.67	3.68	3.73
		3.00	3.55	3.56	3.58	3.49	3.51	3.56
	0.18	0.75	4.29	4.43	4.54	4.18	4.38	4.60
		1.50	3.77	3.81	3.90	3.72	3.82	3.97
		2.25	3.57	3.55	3.55	3.54	3.52	3.58
		3.00	3.53	3.44	3.39	3.44	3.37	3.36
	0.24	0.75	4.15	4.25	4.37	4.22	4.48	4.70
		1.50	3.74	3.66	3.74	3.66	3.77	3.74
		2.25	3.60	3.58	3.50	3.61	3.48	3.54
		3.00	3.52	3.49	3.41	3.52	3.42	3.35

Parameter	Measure	Cross-section S1 f_ck = 30 MPa λ = 120 e₁/h = 0.18			Cross-section S2 f_ck = 30 MPa λ = 120 e₁/h = 0.18			Cross-section S3 f_ck = 30 MPa λ = 120 e₁/h = 0.18
		(1)	(2)	Ratio (2)/(1)	(3)	(4)	Ratio (4)/(3)	(5)	(6)	Ratio (6)/(5)
		ρ = 0.75%	ρ = 3.00%	Ratio (2)/(1)	ρ = 0.75%	ρ = 3.00%	Ratio (4)/(3)	ρ = 0.75%	ρ = 3.00%	Ratio (6)/(5)
Load effect S	μ (kN)	357.8	850.9	2.38	316.0	697.4	2.21	309.5	658.1	2.13
	σ (kN)	36.0	84.3	2.34	31.3	67.4	2.15	32.4	72.3	2.23
	COV	0.10	0.10	0.98	0.10	0.10	0.98	0.10	0.11	1.05
Resistance R	μ (kN)	641.9	1326.5	2.07	590.4	1092.9	1.85	578.5	1050.2	1.82
	σ (kN)	57.9	113.1	1.95	55.5	96.6	1.74	51.4	94.6	1.84
	COV	0.09	0.09	0.95	0.09	0.09	0.94	0.09	0.09	1.01
Safety margin g (R, S)	μ (kN)	284.1	475.6	1.67	274.4	395.5	1.44	269.0	392.1	1.46
	σ (kN)	68.2	141.1	2.07	63.7	117.8	1.85	60.8	119.1	1.96
	COV	0.24	0.30	1.24	0.23	0.30	1.28	0.23	0.30	1.34

[1] Corresponding author: Kleyser Ribeiro. E-mail: kleyser.ribeiro@gmail.com

Brasil

Brasil

Reliability analysis of slender columns using the general method with linear creep theory

Análise da confiabilidade de pilares esbeltos usando o método geral com a teoria linear da fluência

Abstract