Proposition of a modified formulation to predict the lateral-torsional buckling resistance of steel beams: meta-analysis of test data

Pimenta, Roberval José; Leal, Luiz Alberto Araújo de Seixas; Fakury, Ricardo Hallal; Basaglia, Cilmar

doi:10.1590/0370-44672023770134

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Civil Engineering • REM, Int. Eng. J. 77 (4) • Oct-Dec 2024 • https://doi.org/10.1590/0370-44672023770134 copy

Proposition of a modified formulation to predict the lateral-torsional buckling resistance of steel beams: meta-analysis of test data

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Lateral-Torsional Buckling (LTB) is an important limit state that must be considered in the design of steel members subjected to bending. During the process of revising of the Brazilian Standard ABNT NBR 8800:2008, it was questioned whether the present formulation would really lead to good results vis-à-vis the experimental tests. To this end, a comprehensive review of the technical literature was carried out, looking for test results that could be compared with the ones using the equations provided in the Brazilian Standard. Then, a meta-analysis was performed by comparing the test results with the ones of the ABNT NBR 8800:2008 methodology. It was observed that, for beams subjected to uniform bending moments and simple supports, the formulation proved to be adequate. However, for the case of beams with variable bending moments, for which the factor C_b (modification factor for non-uniform bending moment diagram) is used, it was realized that the formulation could lead to unsafe results, especially for high values of C_b. A modified formulation was then proposed, with the introduction of C_b only in the elastic range of the equations (only in the buckling moment equation) and in the equation for the slenderness parameter corresponding to the beginning of yielding, λ_Γ. The proposed modified formulation, as well as the EN 1993-1-1:2022 methodology, were subsequently included in the meta-analysis. Structural reliability analyses were also carried out for both the ABNT NBR 8800:2008 and the modified methodologies. The analyses showed that the reliability indexes of the modified formulation were higher than those of the ABNT NBR 8800:2008 one and closer to the recommended "target value".

Keywords:
lateral torsional buckling; experimental analysis; steel beams; meta-data analysis; LTB

Reference	Shape		C_b	Diagram Shape
Reference	Rolled	Welded	C_b	Diagram Shape
*Lay et al.* (1965)**LAY, M. G.; ADAMS, P. F.; GALAMBOS, T. V. Experiments on high strength steel members. Welding Research Council Bulletin, n. 110, publication n. 287. Lehigh Preserve Institutional Repository, 1965, United States of America.	7	-	1.00
Janss and Massonnet (1967)JANSS, J.; MASSONNET, C. Extension des méthodes de calcul basées sur la plasticité à l'acier A 52. Le déversement plastique – IABSE Publications – Memoires AIPC, Band 27, 1967, ETH-Bibliothek, Germany.	14	-	1.00
	14	-	1.67
Dibley (1969)DIBLEY, J. E. Lateral torsional buckling of I-sections in grade 55 steel. Proceedings of the Institution of Civil Engineers, v. 43, n. 4, p. 599-627, 1969.	30	-	1.00
Kitipornchai and Trahair (1 975)	6	-	1.32
Fukumoto et al. (1980)FUKUMOTO, Y.; ITOH, Y.; KUBO, M. Strength variation of laterally unsupported beams. Journal of the Structural Division, v. 106, n. 1, p. 165-181, 1980.	75	-	1.32
Dux and Kitipornchai (1983)DUX, P. F.; KITIPORNCHAI, S. Inelastic beam buckling experiments. Journal of Constructional Steel Research, v. 3, n. 1, p. 3-9, 1983.	9	-	1.00
			1.14
			1.67
Kubo and Fukumoto (1986)KUBO, M.; FUKUMOTO, Y. Effects of moment distribution on lateral-torsional buckling of rolled steel I-beam. Proceedings of Japan Society of Civil Engineers, 368/I-5, 1986.	44	-	1.32
Wong-Chung and Kitipornchai (1987)WONG-CHUNG, A. D.; KITPORNCHAI, S. Partially braced inelastic beam buckling experiments. Journal of Constructional Steel Research, v. 7, n. 3, p. 189-211, 1987.	11	-	1.00
Foster and Gardner (2015)FOSTER, A. S. J.; GARDNER, L. Ultimate behaviour of steel beams with discrete lateral restraints. Thin-Walled Structures, v. 72, Nov. 2013.	10	-	1.00
	10	-	1.67
Suzuki and Ono (1970)SUZUKI, T.; ONO, T. Experimental study of inelastic beams (3). Transactions of Architectural Institute of Japan (AIJ), n. 175, p. 69-74, 1970.	-	12	1.32
Suzuki and Ono (1973)SUZUKI, T.; ONO, T. Post-lateral buckling of the welded beams. Preprint of Annual Meeting of Architectural Institute of Japan (AIJ), Oct., p. 991-992, 1973.	-	8	1.00
Fukumoto and Itoh (1981)FUKUMOTO, Y.; ITOH, Y. Statistical study of experiments on welded beams. Journal of Structural Division, v. 107, n. 1, p. 89-103, 1981.	-	68	1.32
*Xiong et al.* (2016)**XIONG, G.; KANG, S-B.; YANG, B.; WANG, S.; BAI, J.; NIE, S.; HU, Y.; DAI, G. Experimental and numerical studies on lateral torsional buckling of welded Q460GJ structural steel beams. Engineering Structures, v. 126, Nov., p. 1-14, 2016.	-	8	1.67
Twizel (2021)TWIZELL, S. C. Lateral-torsional buckling response of as-built and heat-straightened welded steel girders. Master’s Thesis. Edmonton: University of Alberta, 2021. 184 f.	-	11	1.14
Total	206	107

Reference	Section Type	C_b	ABNT NBR 8800:2008		Modified Formulation		EN 1993-1-1:2022
Reference	Section Type	C_b	Mean	CoV	Mean	CoV	Mean	CoV
Lsyetal. (1965)	hot-rolled	1.00	1.00	0.02	1.00	0.02	0.95	0.01
Janss and Massonnet (1967)JANSS, J.; MASSONNET, C. Extension des méthodes de calcul basées sur la plasticité à l'acier A 52. Le déversement plastique – IABSE Publications – Memoires AIPC, Band 27, 1967, ETH-Bibliothek, Germany.	hot-rolled	1.00	0.98	0.09	0.98	0.09	0.93	0.09
	hot-rolled	1.67	0.79	0.06	0.79	0.06	0.80	0.07
Dibley(1969)DIBLEY, J. E. Lateral torsional buckling of I-sections in grade 55 steel. Proceedings of the Institution of Civil Engineers, v. 43, n. 4, p. 599-627, 1969.	hot-rolled	1.00	0.98	0.06	0.98	0.06	0.89	0.07
Kitipornchai and Trahair (1975)KITIPORNCHAI, S.; TRAHAIR, N. Inelastic buckling of simply supported steel I-beams. Journal of Structural Division, v. 101, n. 7, p. 1333-1347, 1975.	hot-rolled	1.32	1.10	0.11	0.98	0.04	0.95	0.16
Fukumoto et al. (1980)FUKUMOTO, Y.; ITOH, Y.; KUBO, M. Strength variation of laterally unsupported beams. Journal of the Structural Division, v. 106, n. 1, p. 165-181, 1980.	hot-rolled	1.32	1.14	0.09	0.98	0.08	0.87	0.11
Dux and Kitipornchai (1983)DUX, P. F.; KITIPORNCHAI, S. Inelastic beam buckling experiments. Journal of Constructional Steel Research, v. 3, n. 1, p. 3-9, 1983.	hot-rolled	1.00	1.04	0.03	1.04	0.03	0.95	0.02
		1.14	1.04	0.03	0.95	0.00	0.92	0.00
		1.67	1.03	0.05	0.86	0.01	0.89	0.04
Kubo and Fukumoto (1986)KUBO, M.; FUKUMOTO, Y. Effects of moment distribution on lateral-torsional buckling of rolled steel I-beam. Proceedings of Japan Society of Civil Engineers, 368/I-5, 1986.	hot-rolled	1.32	1.17	011	1.01	0.07	0.88	0.07
Wong-Chung and Kitipornchai (1987)WONG-CHUNG, A. D.; KITPORNCHAI, S. Partially braced inelastic beam buckling experiments. Journal of Constructional Steel Research, v. 7, n. 3, p. 189-211, 1987.	hot-rolled	1.00	1.02	0.02	1.02	0.02	0.86	0.07
Foster and Gardner (2015)FOSTER, A. S. J.; GARDNER, L. Ultimate behaviour of steel beams with discrete lateral restraints. Thin-Walled Structures, v. 72, Nov. 2013.	hot-rolled	1.00	0.97	0.01	0.97	0.01	0.92	0.01
	hot-rolled	1.67	0.96	0.03	0.95	0.03	0.96	0.03
Suzuki and Ono (1970)SUZUKI, T.; ONO, T. Experimental study of inelastic beams (3). Transactions of Architectural Institute of Japan (AIJ), n. 175, p. 69-74, 1970.	welded	1.32	1.04	0.12	0.95	0.06	0.72	0.07
Suzuki and Ono (1973)SUZUKI, T.; ONO, T. Post-lateral buckling of the welded beams. Preprint of Annual Meeting of Architectural Institute of Japan (AIJ), Oct., p. 991-992, 1973.	welded	1.00	1.09	0.01	1.09	0.01	1.06	0.01
Fukumoto and Itoh (1981)FUKUMOTO, Y.; ITOH, Y. Statistical study of experiments on welded beams. Journal of Structural Division, v. 107, n. 1, p. 89-103, 1981.	welded	1.32	1.12	0.15	0.99	0.09	0.88	0.12
Xiong et al. (2016)XIONG, G.; KANG, S-B.; YANG, B.; WANG, S.; BAI, J.; NIE, S.; HU, Y.; DAI, G. Experimental and numerical studies on lateral torsional buckling of welded Q460GJ structural steel beams. Engineering Structures, v. 126, Nov., p. 1-14, 2016.	welded	1.67	1.10	0.09	0.92	0.03	0.86	0.06
Twizel (2021)TWIZELL, S. C. Lateral-torsional buckling response of as-built and heat-straightened welded steel girders. Master’s Thesis. Edmonton: University of Alberta, 2021. 184 f.	welded	1.14	0.91	0.09	0.90	0.08	0.61	0.05

Parameters
Variables		δ	CoV	Distribution	Reference
Load	Dead D	1.06	0.120	Normal	*Santiago et al.* (2019)**SANTIAGO, W. C.; KROETZ, H. M.; BECK, A.T. Reliability-based calibration of Brazilian structural design codes used in the design of concrete structures, IBRACON Structures and Materials Journal, v. 12, n. 6, p. 1288-1304, 2019.
Load	Live L	0.93	0.250	Gumbel	*Costa et aí.* (2023)**COSTA, L. G. L.; SANTIAGO, W. C.; BECK, A. T. Probabilistic models for live loads in buildings: critical review, comparison to brazilian design standards and calibration of partial safety factors. IBRACON Structures and Materials Journal, v. 16, n. 2, e16204, 2023.
Resistance	Plastic Modulus Ζ_x (RS)	1.00	0.040	Normal	JCSS(2001)JOINT COMMITTEE ON STRUCTURAL SAFETY. Probabilistic model code – Parts 1 to 3, 2001-2002, European Community.
	Geometric K_t (RS)	1.00	0.040	Normal
	Plastic Modulus Ζ_χ (WS)	1.03	0.023	Normal	Pimenta (2008)PIMENTA, R. J. Perfis de alma senoidal: proposição de métodos de cálculo e análise da confiabilidade estrutural. Tese (Doutorado) - Universidade Federal de Minas Gerais, Belo Horizonte, 2008. 247 f.
	Geometric K_t (RS)	1.05	0.030	Normal
	Yield Strength F (RS)	1.11	0.055	Lognormal	This work
	Yield Strength F (WS)	1.17	0.065	Lognormal	Pimenta (2008)PIMENTA, R. J. Perfis de alma senoidal: proposição de métodos de cálculo e análise da confiabilidade estrutural. Tese (Doutorado) - Universidade Federal de Minas Gerais, Belo Horizonte, 2008. 247 f.
	Residual Stress Σ_Γ	1.00	0.300	Normal
	Modulus of Elasticity Ε	1.03	0.022	Lognormal

All - Plastic			All- Inelastic
q	β		q	β
q	ABNT NBR 8800:2008	Modified formulation	q	ABNT NBR 8800:2008	Modified formulation
0.5	3.1	3.1	0.5	2.3	3.2
1.0	3.0	3.0	1.0	2.3	3.0
1.5	2.9	2.9	1.5	2.3	2.9
2.0	2.8	2.8	2.0	2.3	2.9
2.5	2.8	2.8	2.5	2.2	2.8
3.0	2.8	2.8	3.0	2.2	2.8
3.5	2.7	2.7	3.5	2.2	2.7
4.0	2.7	2.7	4.0	2.2	2.7

Professional coefficient Ρ			Professional coefficient Ρ
Parameters	ABNT NBR 8800:2008	Modified formulation	Parameters	ABNT NBR 8800:2008	Modified formulation
δ	1.047	1.047	δ	0.896	1.020
CoV	0.112	0.112	CoV	0.111	0.091

Rolled - Inelastic			Welded - Inelastic
q	β		q	β
q	NBR 8800:2008	Modified formulation	q	NBR 8800:2008	Modified formulation
0.5	2.4	3.3	0.5	2.7	3.5
1.0	2.4	3.1	1.0	2.6	3.3
1.5	2.4	3.0	1.5	2.5	3.2
2.0	2.3	2.9	2.0	2.5	3.1
2.5	2.3	2.8	2.5	2.4	3.0
3.0	2.3	2.8	3.0	2.4	3.0
3.5	2.3	2.8	3.5	2.4	3.0
4.0	2.3	2.7	4.0	2.4	3.0

Professional coefficient Ρ			Professional coefficient Ρ
Parameters	NBR 8800:2008	Modified formulation	Parameters	NBR 8800:2008	Modified formulation
δ	0.913	1.024	δ	0.852	1.012
CoV	0.109	0.084	CoV	0.099	0.104

All-Inelastic-C_b = 1.0			All - Inelastic-C_b > 1.0
q	β		q	β
q	ABNT NBR 8800:2008	Modified formulation	q	ABNT NBR 8800:2008	Modified formulation
0.5	3.0	3.0	0.5	2.2	3.3
1.0	2.9	2.9	1.0	2.3	3.1
1.5	2.8	2.8	1.5	2.2	3.0
2.0	2.7	2.7	2.0	2.2	2.9
2.5	2.6	2.6	2.5	2.2	2.8
3.0	2.6	2.6	3.0	2.2	2.8
3.5	2.6	2.6	3.5	2.1	2.8
4.0	2.6	2.6	4.0	2.1	2.7

Professional coefficient P			Professional coefficient P
Parameters	NBR 8800:2008	Modified formulation	Parameters	NBR 8800:2008	Modified formulation
δ	0.973	0.973	δ	0.877	1.029
CoV	0.087	0.087	CoV	0.107	0.089

All - Elastic
q	β
q	ABNT NBR 8800:2008	Modified formulation
0.5	2.9	3.5
1.0	2.8	3.2
1.5	2.7	3.0
2.0	2.6	2.9
2.5	2.6	2.9
3.0	2.5	2.8
3.5	2.5	2.8
4.0	2.5	2.8

Professional coefficient Ρ
Parameters	ABNT NBR 8800:2008	Modified formulation
δ	1.027	1.089
CoV	0.106	0.085

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