Minimum squares method in the use of rotation moment curves in the analysis of semi-rigited plane frames

Silva, Everton Luis Consoni da; Souza, Luiz Antonio Farani de; Meirelles, Paulo Henrique de Freitas; Wutzow, Wilson Wesley

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Artigos • Matéria (Rio J.) 28 (1) • 2023 • https://doi.org/10.1590/1517-7076-RMAT-2022-0285 copy

Minimum squares method in the use of rotation moment curves in the analysis of semi-rigited plane frames

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Usually, in the development of projects, it is assumed that the behavior of the connections between the beam-column of the system is idealized through two extremes: the flexible connections in which no moment is transmitted between the column and the beam and these elements are behave independently; and the fully rigid connections in which full moment transmission occurs. However, in real structures, experimental investigations have shown that most connections should be treated as semi-rigid connections. This behavior is described by means of moment-rotation curves, which, in turn, must be incorporated into the structural analysis in order to obtain more accurate information about the performance of the connection. Therefore, this article presents the application of the method of Ordinary Least Squares for the simulation of the semi-rigid behavior of connections of metallic plane optics. Through coupling with a numerical-computational model for prediction of frames with geometric nonlinear behavior and semi-rigid connections. For this purpose, the Corrotational Finite Element formulation is used, considering the Euler-Bernoulli beam theory. The connection between the structural members is simulated by a hybrid element. The system of nonlinear equations is solved by the incremental-iterative procedure of Potra Pták, associated with the Cylindrical Arc Length technique. Four structures found in the literature are analyzed, when compared, it is possible to affirm that the proposed method presented excellent results, faithfully representing the equilibrium path of the structure considering the deterioration of the connection.

Keywords
Semi-rigid connections; Corrotational Formulation; Potra-Pták Method; Co-rotational formulation

POLINÔMIO INTERPOLADOR
M = –0,07863 θ_r ⁸+ 2,054 θ_r ⁷ – 22,64 θ_r ⁶+ 137 θ_r ⁵ – 497,3 θ_r ⁴+ 1109 θ_r ³ – 1504 θ_r ²+ 1200 θ_r – 4,96

CURVAS	kt	Np	kt/Np	t(s)
Rígido (u/l)	46	17	2,7059	0,132
Semirrígido (u/l)	85	27	3,1481	0,214
Rígido 1 – (v/l)	46	17	2,7059	0,126
Semirrígida – Proposta 1 – (v/l)	87	27	3,1481	0,187

CURVA	POLINÔMIO INTERPOLADOR
A	$M = - 2, 612 \times 10^{16} θ_{r}^{8} + 5, 949 \times 10^{15} θ_{r}^{7} - 5, 532 \times 10^{14} θ_{r}^{6} + 2, 711 \times 10^{13} θ_{r}^{5} - 7, 545 \times 10^{11} θ_{r}^{4} + 1, 198 \times 10^{10} θ_{r}^{3} - 1, 037 \times 10^{8} θ_{r}^{2} + 4, 694 \times 10^{5} θ_{r} - 2, 794 \times 10^{- 9}$
B	$M = - 8, 563 \times 10^{15} θ_{r}^{8} + 2, 368 \times 10^{15} θ_{r}^{7} - 2, 732 \times 10^{14} θ_{r}^{6} + 1, 697 \times 10^{13} θ_{r}^{5} - 6, 122 \times 10^{11} θ_{r}^{4} + 1, 29 \times 10^{10} θ_{r}^{3} - 1, 552 \times 10^{8} θ_{r}^{2} + 1, 092 \times 10^{6} θ_{r} - 8, 225$
C	$M = 2, 869 \times 10^{15} θ_{r}^{8} - 4, 841 \times 10^{14} θ_{r}^{7} + 1, 797 \times 10^{13} θ_{r}^{6} + 1, 398 \times 10^{12} θ_{r}^{5} - 1, 507 \times 10^{11} θ_{r}^{4} + 5, 763 \times 10^{9} θ_{r}^{3} - 1, 165 \times 10^{8} θ_{r}^{2} + 1, 418 \times 10^{6} θ_{r} - 75, 92$
D	$M = 7, 243 \times 10^{16} θ_{r}^{8} - 1, 724 \times 10^{16} θ_{r}^{7} + 1, 646 \times 10^{15} θ_{r}^{6} - 7, 927 \times 10^{13} θ_{r}^{5} + 1, 928 \times 10^{12} θ_{r}^{4} - 1, 688 \times 10^{10} θ_{r}^{3} - 1, 789 \times 10^{8} θ_{r}^{2} + 4, 594 \times 10^{6} θ_{r} - 419, 7$

CURVAS	CHEN e LUI [⁵[5] CHEN, W.-F., LUI, E., Stability design of steel frames, London, Boca Raton, 1991.]	CHAN e CHUI [³¹[31] CHAN, S.L., CHUI, P.P.T., Non-linear static and cyclic analysis of steel frames with semi-rigid connections, Elsevier, USA, 2000.]	TRABALHO ATUAL
A	400,3	395,9	396,08
B	911,9	907,4	899,5
C	1076,5	1107,6	1096,35
D	2112,9	2121,8	2119.82
Ligação Rígida	3225,0	3318,4	3302,48

CURVAS	kt	Np	Kt/Np	t(s)
A	28	9	3,111	0,5311
B	37	12	3,083	0,7342
C	55	17	3,235	0,8762
D	72	22	3,273	1,4746
Ligação Rígida	181	89	2,034	2,9055

CURVAS	CHEN E LUI [⁵[5] CHEN, W.-F., LUI, E., Stability design of steel frames, London, Boca Raton, 1991.]	CHAN E CHUI [³¹[31] CHAN, S.L., CHUI, P.P.T., Non-linear static and cyclic analysis of steel frames with semi-rigid connections, Elsevier, USA, 2000.]	TRABALHO ATUAL
A	2802,4	2838	2861,87
B	3825,5	3892,2	3857,27
C	4581,7	4617,3	4628,07
D	7228,4	7330,7	7297,18
Ligação Rígida	11254	11387,5	11400,39

CURVAS	kt	Np	Kt/Np	t(s)
A	18	5	3,6	0,2284
B	34	9	3,777	0,7889
C	55	17	3,235	0,8762
D	120	23	5,217	1,4746
Ligação Rígida	135	63	2,143	1,6725

CURVA	POLINÔMIO INTERPOLADOR
A	$M = - 5, 165 \times 10^{15} θ_{r}^{8} + 1, 109 \times 10^{15} θ_{r}^{7} - 9, 858 \times 10^{13} θ_{r}^{6} + 4, 694 \times 10^{12} θ_{r}^{5} - 1, 294 \times 10^{11} θ_{r}^{4} + 2, 096 \times 10^{9} θ_{r}^{3} - 1, 951 \times 10^{7} θ_{r}^{2} + 1, 039 \times 10^{5} θ_{r} + 3, 463$
B	$M = 1, 523 \times 10^{14} θ_{r}^{8} - 3, 267 \times 10^{13} θ_{r}^{7} + 2, 829 \times 10^{12} θ_{r}^{6} - 1, 238 \times 10^{11} θ_{r}^{5} + 2, 664 \times 10^{9} θ_{r}^{4} - 1, 323 \times 10^{7} θ_{r}^{3} - 5, 875 \times 10^{5} θ_{r}^{2} + 1, 274 \times 10^{4} θ_{r} - 0, 2053$

CURVAS	NEFOVSKA-DANILOVIĆ E SEKULOVIĆ [³¹[31] CHAN, S.L., CHUI, P.P.T., Non-linear static and cyclic analysis of steel frames with semi-rigid connections, Elsevier, USA, 2000.]	TRABALHO ATUAL
Curva A	1878,33	1853,34
Modelo linear – A	2177,75	2175,89
Curva B	940,24	937,79
Modelo linear – B	1044,07	1044,04
Ligação Rígida	2742,43	2722,95

CURVAS	kt	Np	kt/Np	t(s)
Rígido	276	66	4,1818	13,5149
Curva A	58	11	5,273	2,7874
Modelo linear – A	39	11	3,5454	1,6832
Curva B	40	9	4,444	2,7495
Modelo linear – B	32	9	3,555	1,5738

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E-mail: revmateria@gmail.com

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[1] e-mail: pg403386@uem.br, lasouza@utfpr.edu.br, pg403395@uem.br, wwwutzow@uem.br