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A stabilized smoothed finite element method for free vibration analysis of Mindlin-Reissner plates. (English) Zbl 1172.74047
Summary: We perform a free vibration analysis of Mindlin-Reissner plates using the stabilized smoothed finite element method. The bending strains of the MITC4 and STAB elements are incorporated with a cell-wise smoothing operation to give new proposed elements, the mixed interpolation and smoothed curvature (MISC\(k\)) and SMISC\(k\) elements. The corresponding bending stiffness matrix is computed along the boundaries of the smoothing elements (smoothing cells). Note that shearing strains and the shearing stiffness matrix of the proposed elements are the same as for the original MITC4 and STAB elements. It is confirmed by numerical tests that the present method is free of shear locking and has marginal improvements compared with the original elements.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
Software:
Mfree2D
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References:
[1] Zienkiewicz, Reduced integration technique in general analysis of plates and shells. Simple and efficient element for plate bending, International Journal for Numerical Methods in Engineering 3 pp 275– (1971)
[2] Hughes, Simple and efficient element for plate bending, International Journal for Numerical Methods in Engineering 11 pp 1529– (1977) · Zbl 0363.73067
[3] Hughes, Reduced and selective integration techniques in finite element method of plates, Nuclear Engineering Design 46 pp 203– (1978)
[4] Lee, Finite elements based upon Mindlin plate theory with particular reference to the four-node isoparametric element, AIAA Journal 16 pp 29– (1978)
[5] Lee, Mixed formulation finite elements for Mindlin theory plate bending, International Journal for Numerical Methods in Engineering 18 pp 1297– (1982) · Zbl 0486.73069
[6] Geyer, On reduced integration and locking of flat shell finite elements with drilling rotations, Communications in Numerical Methods in Engineering 19 pp 85– (2003) · Zbl 1024.74040
[7] Luo, An accurate quadrilateral plate element based on energy optimization, Communications in Numerical Methods in Engineering 21 pp 487– (2005) · Zbl 1084.74057
[8] Hughes, Finite elements based upon Mindlin plate theory with particular reference to the four-node isoparametric element, Journal of Applied Mechanics 48 pp 587– (1981) · Zbl 0459.73069
[9] Bathe, A four-node plate bending element based on Mindlin/Reissener plate theory and a mixed interpolation, International Journal for Numerical Methods in Engineering 21 pp 367– (1985) · Zbl 0551.73072
[10] Bathe, A formulation of general shell elements. The use of mixed interpolation of tensorial components, International Journal for Numerical Methods in Engineering 22 pp 697– (1986) · Zbl 0585.73123
[11] Dvorkin, A continuum mechanics based four-node shell element for general nonlinear analysis, Engineering Computations 1 pp 77– (1984)
[12] Simo, On the variational foundation of assumed strain methods, Journal of Applied Mechanics 53 pp 51– (1986) · Zbl 0592.73019
[13] Simo, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 29 pp 1595– (1990) · Zbl 0724.73222
[14] Piltner, A mixed finite element for plate bending with eight enhanced strain modes, Communications in Numerical Methods in Engineering 17 pp 443– (2001) · Zbl 1014.74072
[15] Bathe, Finite Element Procedures (1996)
[16] Zienkiewicz, The Finite Element Method (2000) · Zbl 0962.76056
[17] Liu, A smoothed finite element for mechanics problems, Computational Mechanics 39 pp 859– (2007) · Zbl 1169.74047
[18] Chen, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081
[19] Wang, Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation, Computer Methods in Applied Mechanics and Engineering 193 pp 1065– (2004) · Zbl 1060.74675
[20] Wang, A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration, International Journal for Numerical Methods in Engineering (2008) · Zbl 1159.74460
[21] Nguyen-Xuan, A smoothed finite element method for plate analysis, Computer Methods in Applied Mechanics and Engineering 197 pp 1184– (2008) · Zbl 1159.74434
[22] Liu, Theoretical aspects of the smoothed finite element method (SFEM), International Journal for Numerical Methods in Engineering 71 pp 902– (2007) · Zbl 1194.74432
[23] Nguyen, Selective smoothed finite element method, Tsinghua Science and Technology 12 pp 497– (2007)
[24] Dai, Free and forced vibration analysis using the smoothed finite element method (SFEM), Journal of Sound and Vibration 301 pp 803– (2007)
[25] Batoz, Modélisation des Structures par Éléments Finis (Vol. 2): Poutres et Plaques (1990)
[26] Thompson, On optimal stabilized MITC4 plate bending elements for accurate frequency response analysis, Computers and Structures 81 pp 995– (2003)
[27] Lyly, A stable bilinear element for Reissner-Mindlin plate model, Computer Methods in Applied Mechanics and Engineering 110 pp 243– (1993) · Zbl 0846.73065
[28] Abbassian, Free Vibration Benchmarks pp 40– (1987)
[29] Robert, Formulas for Natural Frequency and Mode Shape (1979)
[30] Liu, Mesh-free Methods: Moving Beyond the Finite Element Method (2002)
[31] Karunasena, Natural frequencies of thick arbitrary quadrilateral plates using the pb-2 Ritz method, Journal of Sound and Vibration 196 (4) pp 371– (1996)
[32] Gorman, Free vibration analysis of cantilever plates with step discontinuities in properties by the method of superposition, Journal of Sound and Vibration 253 pp 631– (2002)
[33] Xiang, Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundation, International Journal of Mechanical Sciences 45 pp 1229– (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.