An Explicit Consistent Geometric Stiffness Matrix for the DKT Element

Lucena, Eliseu; Araripe, Marcus Antônio Ferreira; Monteiro, Francisco Alex Correia; Hernandes, José Antônio

doi:10.1590/1679-78252799

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Latin American Journal of Solids and Structures

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Articles • Lat. Am. j. solids struct. 14 (4) • Mar 2017 • https://doi.org/10.1590/1679-78252799 copy

An Explicit Consistent Geometric Stiffness Matrix for the DKT Element

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

A large number of references dealing with the geometric stiffness matrix of the DKT finite element exist in the literature, where nearly all of them adopt an inconsistent form. While such a matrix may be part of the element to treat nonlinear problems in general, it is of crucial importance for linearized buckling analysis. The present work seems to be the first to obtain an explicit expression for this matrix in a consistent way. Numerical results on linear buckling of plates assess the element performance either with the proposed explicit consistent matrix, or with the most commonly used inconsistent matrix.

Keywords:
DKT; geometric stiffness matrix; discrete Kirchhoff triangle; buckling

		_^kx /_^pxy = 0) (			_^kxy / ^a
	mesh	_^py = 0	_^py = _^px	_^py = -_^px	_^px = _^py = 0
isotropic	1×1	1.119^b	1.121	1.231	-
		0.903^c	0.903	∞	-
	2×2	1.016	1.020	1.003	1.546
		0.992	0.992	1.167	0.422
	4×4	1.003	1.004	0.994	1.007
		0.998	0.998	1.041	0.813
	8×8	1.001	1.001	0.997	0.982
		0.999	0.999	1.010	0.954
	16×16	1.000	1.000	0.999	0.994
		1.000	1.000	1.003	0.988

		4.000^d	2.000^d	8.333^d	9.325^e
orthotropic	1×1	1.085	1.097	1.264	-
		0.837	0.837	∞	-
	2×2	1.016	1.012	0.996	1.649
		0.973	0.969	1.245	0.449
	4×4	1.003	1.001	0.989	1.029
		0.991	0.991	1.056	0.826
	8×8	1.001	1.000	0.995	0.986
		0.998	0.997	1.013	0.950
	16×16	1.000	1.000	0.999	0.999
		0.999	0.999	1.003	0.986

		13.000^d	6.500^d	-11.333^d	28.066^e

		_^kx /_^pxy = 0) (			_^kxy / ^a
	mesh	_^py = 0	_^py = _^px	_^py = -_^px	_^px = _^py = 0
isotropic	1×1	1.092^b	1.140	0.951	-
		∞^c	0.743	∞	-
	2×2	1.004	1.055	0.913	1.385
		1.054	1.032	1.071	0.487
	4×4	0.995	1.014	0.958	0.966
		1.014	1.013	1.010	0.854
	8×8	0.998	1.003	0.988	0.976
		1.003	1.003	1.001	0.964
	16×16	1.000	1.001	0.997	0.993
		1.001	1.001	1.000	0.991

		7.691^d	3.830^d	10.788^d	12.565^e
orthotropic	1×1	1.338	1.416	1.072	-
		0.739	0.846	∞	-
	2×2	1.182	1.104	1.395	1.858
		1.089	1.077	2.120	0.497
	4×4	1.038	1.020	1.044	1.114
		1.020	1.023	1.179	0.864
	8×8	1.008	1.004	1.008	1.016
		1.004	1.004	1.039	0.958
	16×16	1.002	1.001	1.002	1.000
		1.001	1.001	1.009	0.988

		17.604^d	7.693^d	15.169^d	29.567^e

		_^kx /_^pxy = 0) (			_^kxy / ^a
	mesh	_^py = 0	_^py = _^px	_^py = -_^px	_^px = _^py = 0
isotropic	1×1	2.002^b	1.920	1.087	-
		0.809^c	0.768	∞	-
	2×2	1.055	1.099	1.096	1.326
		1.043	1.062	1.340	0.583
	4×4	1.010	1.019	0.989	1.000
		1.018	1.019	1.080	0.861
	8×8	1.002	1.004	0.994	0.980
		1.005	1.005	1.018	0.962
	16×16	1.000	1.001	0.998	0.993
		1.001	1.001	1.004	0.989

		10.074^d	5.304^d	14.966^d	14.642^d
orthotropic	1×1	1.622	0.975	∞	-
		0.649	∞	∞	-
	2×2	1.094	1.247	1.160	1.257
		0.992	1.433	1.237	0.768
	4×4	1.016	1.016	1.015	1.137
		1.002	1.060	1.191	0.909
	8×8	1.004	1.002	0.999	1.014
		1.000	1.012	1.042	0.947
	16×16	1.001	1.000	0.999	0.994
		1.000	1.003	1.010	0.983

		46.289^d	16.681^d	24.066^d	46.726^d

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[1] * Corresponding author: eliseu@ita.br