Revisiting Some Developments of Boundary Elements for Thick Plates in Brazil

Marczak, Rogério José

doi:10.1590/1679-78251441

Rogério José Marczak

Mechanical Engineering Dept. - UFRGS. Rua Sarmento Leite 425, Porto Alegre, Brazil - 90050-170 rato@mecanica.ufrgs.br

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Figure 1
General algorithm to solve non-linear problems described by eqs. (14-17).

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Figure 2
Illustration of the composition of two-dimensional domain partitions using geometric and physical partitions.

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Figure 3
Code fragment of the solveCurrentStep member for linear applications.

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Figure 4
Code fragment of the solveCurrentStep member for non-linear applications.

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Figure 5
Code fragment of the formBVPTangent member for linear static applications.

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Figure 6
The two major hierarchy levels of the mcAnalysis class.

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Figure 7
Minimal class diagram of a complete application code.

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Figure 8
Code for a BEM linear elastic analysis.

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Figure 9
Code for a BEM transient dynamic analysis.

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Figure 10
Code excerpt showing the integration and assembly of typical BEM matrices.

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Figure 11
Code showing a more elaborate integration procedure to handle different singularities in BEM matrices.

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Figure 12
p-convergence rates for plates under uniform loading. (a) square plate. (b) circular plate.

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Figure 13
Comparison of convergence curves for several numerical formulations. Clamped square plate under uniform loading. Solid symbols refer to the present results.

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Figure 14
Convergence curves for thin square plates under uniform loading using RBM technique, Kutt's quadrature or the direct method to integrate singular kernels. (a) Constant BE. (b) Linear BE. (c) Quadratic BE.

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Figure 15
Convergence curves for thick square plates under uniform loading using the direct method to integrate singular kernels. (a) h/a = 0.15. (b) h/a = 0.20.

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Figure 16
Shear force along the side of a clamped, uniformly loaded square plate. The continuous line refers do the analytical solution and the symbols to the present results.

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Figure 17
First buckling mode of a clamped square plate under pure shear. Result generated using 6×6 constant domain cells inside the domain, and 26 constant boundary elements.

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Figure 18
Two subregion non-linear plate bending example under uniform load.

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Figure 19
Non-linear results for non-null rotation along the central line for the case illustrated in .

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Table 1
Numerical results for clamped square plates under uniform loading.

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Table 2
Numerical results for SS-1 supported square plates under uniform loading.

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Table 3
Numerical results for SS2-supported square plates under uniform loading.

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Table 4
Numerical results for critical load factor of supported square plates under axial compression. Constant boundary elements and domain cells.

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Table 5
Numerical results for critical load factor of supported square plates under biaxial compression. Constant boundary elements and domain cells.

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Table 6
Numerical results for non-linear bending of SS1 square plates under uniform load.

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Table 7
Numerical results for non-linear bending of SS2 square plates under uniform load.

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			h/a
	0.05	0.10	0.15	0.20	0.25
Yuan & Miller[yuan88]	0.01417	0.01618	0.01939	0.02371	0.02913
Deshmukh & Archer[des73]	0.01451	0.01643	-	0.02366	-
Present - Linear (6×6 mesh)	0.01453	0.01648	0.01957	0.02378	0.02009
Present - Quadratic (3×3 mesh)	0.01451	0.01645	0.01953	0.02373	0.02904

			h/a
	0.05	0.10	0.15	0.20	0.25
Yuan & Miller[yuan88]	0.04650	0.05019	0.05480	0.06018	0.06683
Deshmukh & Archer[des73]	0.04677	0.05009	-	0.05900	-
Present - Linear (6×6 mesh)	0.04535	0.04882	0.05367	0.05954	0.06631
Present - Quadratic (3×3 mesh)	0.04578	0.04942	0.05432	0.06014	0.06684

		h/a
	0.05	0.10	0.15	0.20
Salerno & Goldberg [1960]Salerno, V.L. & Goldberg, M.A., (1960). Effect of Shear Deformation on the Bending of Rectangular Plates, Journal of Applied Mechanics, pp.27:54--58. (Mindlin)	0.04486	0.04632	0.04676	0.05360
Wang et al. [wang01] (Reissner)	0.04488	0.04630	0.04870	0.05220
Lee et al. [lee01] (Mindlin)	0.04488	0.04663	0.04958	0.05355
Kant & Hinton [kant80] (Higher order model)	--	0.04663	--	0.05351
Craig [craig87] (Reissner)	0.04492	0.04683	0.05183	0.05353
Katsikadelis & Yotis [kats93] (Reissner)	0.04487	0.04634	--	0.05221
Long et al.[long88] (Reissner)	0.04458	0.04612	--	0.05220
Present - Linear (6×6 mesh)	0.04493	0.04668	0.04959	0.05366
Present - Quadratic (3×3 mesh)	0.04497	0.04672	0.04961	0.05368

		h/a
	0.001	0.01	0.05	0.1
3D elasticity (Dawe & Roufaeil, 1982Dawe, D.J. & Roufaeil, O.L., (1982). Buckling of Rectangular Mindlin Plates, Computers & Structures 15, pp.461--471.)	4.000	-	3.911	3.741
Rayleigh-Ritz (Dawe & Roufaeil, 1982Dawe, D.J. & Roufaeil, O.L., (1982). Buckling of Rectangular Mindlin Plates, Computers & Structures 15, pp.461--471.)	4.000	-	3.929	3.731
FE - 9LE (Cheung et al., 1986Cheung, Y.K., Chan, A.H.C. & Tham, L.G., (1986). A Study on the Linear Elastic Stability of Mindlin Plates, Int. J. Numer. Methods Eng. 22, pp.117--132.) - 8×8 mesh	-	4.100	-	3.758
FE - 36LE (Cheung et al., 1986Cheung, Y.K., Chan, A.H.C. & Tham, L.G., (1986). A Study on the Linear Elastic Stability of Mindlin Plates, Int. J. Numer. Methods Eng. 22, pp.117--132.) - 2×2 mesh	-	3.998	-	3.732
BE - Present work (6×6 mesh)	3.9671	3.9646	3.8977	3.7694

	h/a
	0.01	0.1
FE - 8LE (Cheung et al., 1986Cheung, Y.K., Chan, A.H.C. & Tham, L.G., (1986). A Study on the Linear Elastic Stability of Mindlin Plates, Int. J. Numer. Methods Eng. 22, pp.117--132.) - 8×8 mesh	2.030	1.880
FE - 17LE (Cheung et al., 1986Cheung, Y.K., Chan, A.H.C. & Tham, L.G., (1986). A Study on the Linear Elastic Stability of Mindlin Plates, Int. J. Numer. Methods Eng. 22, pp.117--132.) - 3×3 mesh	2.000	1.870
BE - Present work (6(6 mesh)	1.9841	1.8818

		R
	0.9158	4.579	6.868	9.158
Rayleigh-Ritz (Azizian & Dawe, 1985Azizian, Z. G. & Dawe, D. J., (1985). Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method, Computers & Structures 21, pp.423--436.)	0.04053	0.1929	0.2750	0.3467
FSM (Azizian & Dawe, 1985Azizian, Z. G. & Dawe, D. J., (1985). Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method, Computers & Structures 21, pp.423--436.)	0.04205	0.1950	0.2776	0.3494
BEM (Xiao-Yan et al., 1990Xiao-Yan, L., Mao-Kwang, H. & Xiuxi, W., (1990), Geometrically Nonlinear Analysis of a Reissner Type Plate by the Boundary Element Method, Computers & Structures 37, pp.911--916.)	0.04090	0.1942	0.2767	0.3489
Present work (SS2) BE1/DC0	0.04200	0.1969	0.2775	0.3464
Present work (SS2) BE1/DC1	0.04199	0.1958	0.2753	0.3425
Present work (SS1) BE1/DC0	0.04428	0.2056	0.2878	0.3573
Present work (SS1) BE1/DC1	0.04426	0.2041	0.2849	0.3534

		R
	0.9158	4.579	6.868	9.158
Rayleigh-Ritz (Azizian & Dawe, 1985Azizian, Z. G. & Dawe, D. J., (1985). Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method, Computers & Structures 21, pp.423--436.)	0.01915	0.09513	0.1416	0.1867
FSM (Xiao-Yan et al., 1990Xiao-Yan, L., Mao-Kwang, H. & Xiuxi, W., (1990), Geometrically Nonlinear Analysis of a Reissner Type Plate by the Boundary Element Method, Computers & Structures 37, pp.911--916.)	0.01991	0.09883	0.1469	0.1936
BEM [xiao90]	0.01991	0.09840	0.1455	0.1904
Present work BE1/DC0 x	0.1002	0.1002	0.1488	0.1957
Present work BE1/DC1	0.02020	0.1001	0.1485	0.1952

Brasil

Brasil

Abstract