# A cell-based smoothed three-node plate finite element with a bubble node for static analyses of both thin and thick plates

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https://doi.org/10.15625/0866-7136/8809## Keywords:

shear locking, MITC3 , cell-based smoothed technique, CS-MITC3 plate elements## Abstract

This paper develops the cell-based (CS) smoothed finite element method for a three-node plate finite element with a bubble node at the centroid of the element. Based on the first-order shear deformation theory, the in-plane strains are smoothed on three non-overlapped subdomains of the element to transform the numerical integration of the element stiffness matrix from the surfaces into the lines of the subdomains. The shear-locking phenomenon, which occurs when the plate's thickness becomes small, is removed by employing the mixed interpolation of tensorial components (MITC). The present element, namely CS-MITC3+, passes the patch test and behaves independently from the sequence of node numbers of the element. Numerical results given by the CS-MITC3+ elements are better than the MITC3+ elements. As compared to other smoothed three-node plate finite elements, the CS-MITC3+ is a good competitor.

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T. J. R. Hughes, M. Cohen, and M. Haroun. Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engineering and Design, 46, (1), (1978), pp. 203– 222. doi:10.1016/0029-5493(78)90184-x.

J. L. Batoz, K. J. Bathe, and L. W. Ho. A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, 15, (12), (1980), pp. 1771–1812. doi:10.1002/nme.1620151205.

R. H. Macneal. Derivation of element stiffness matrices by assumed strain distributions. Nuclear Engineering and Design, 70, (1), (1982), pp. 3–12. doi:10.1016/0029-5493(82)90262-x.

A. Tessler and T. J. R. Hughes. A three-node Mindlin plate element with improved transverse shear. Computer Methods in Applied Mechanics and Engineering, 50, (1), (1985), pp. 71–101. doi:10.1016/0045-7825(85)90114-8.

U. Andelfinger and E. Ramm. EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. International Journal for Numerical Methods in Engineering, 36, (8), (1993), pp. 1311–1337. doi:10.1002/nme.1620360805.

K. U. Bletzinger, M. Bischoff, and E. Ramm. A unified approach for shear-locking-free triangular and rectangular shell finite elements. Computers & Structures, 75, (3), (2000), pp. 321– 334. doi:10.1016/s0045-7949(99)00140-6.

K. J. Bathe. Finite element procedures. Prentice Hall International, Inc, (1996).

K. J. Bathe and E. N. Dvorkin. A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. International Journal for Numerical Methods in Engineering, 21, (2), (1985), pp. 367–383. doi:10.1002/nme.1620210213.

K. J. Bathe and E. N. Dvorkin. A formulation of general shell elements–the use of mixed interpolation of tensorial components. International Journal for Numerical Methods in Engineering, 22, (3), (1986), pp. 697–722. doi:10.1002/nme.1620220312.

M. L. Bucalem and K. J. Bathe. Higher-order MITC general shell elements. International Journal for Numerical Methods in Engineering, 36, (21), (1993), pp. 3729–3754. doi:10.1002/nme.1620362109.

P. S. Lee and K. J. Bathe. Development of MITC isotropic triangular shell finite elements.

Computers & Structures, 82, (11), (2004), pp. 945–962. doi:10.1016/j.compstruc.2004.02.004.

K. J. Bathe, F. Brezzi, and S. W. Cho. The MITC7 and MITC9 plate bending elements. Computers & Structures, 32, (3-4), (1989), pp. 797–814. doi:10.1016/0045-7949(89)90365-9.

Y. Ko, P. S. Lee, and K. J. Bathe. The MITC4+ shell element and its performance. Computers & Structures, 169, (2016), pp. 57–68. doi:10.1016/j.compstruc.2016.03.002.

Y. Lee, P. S. Lee, and K. J. Bathe. The MITC3+ shell element and its performance. Computers & Structures, 138, (2014), pp. 12–23. doi:10.1016/j.compstruc.2014.02.005.

G. R. Liu, K. Y. Dai, and T. T. Nguyen. A smoothed finite element method for mechanics problems. Computational Mechanics, 39, (6), (2007), pp. 859–877. doi:10.1007/s00466-006-0075-4.

G. R. Liu and N. T. Trung. Smoothed finite element methods. CRC press, (2016). doi:10.1201/ebk1439820278.

H. Nguyen-Xuan and G. R. Liu. An edge-based smoothed finite element method softened with a bubble function (bES-FEM) for solid mechanics problems. Computers & Structures, 128, (2013), pp. 14–30. doi:10.1016/j.compstruc.2013.05.009.

G. R. Liu, T. Nguyen-Thoi, and K. Y. Lam. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. Journal of Sound and Vibration, 320, (4), (2009), pp. 1100–1130. doi:10.1016/j.jsv.2008.08.027.

H. Nguyen-Xuan, T. Rabczuk, S. Bordas, and J. F. Debongnie. A smoothed finite element method for plate analysis. Computer Methods in Applied Mechanics and Engineering, 197, (13), (2008), pp. 1184–1203. doi:10.1016/j.cma.2007.10.008.

T. Nguyen-Thoi, P. Phung-Van, H. Luong-Van, H. Nguyen-Van, and H. Nguyen-Xuan. A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates. Computational Mechanics, 51, (1), (2013), pp. 65–81. doi:10.1007/s00466-012-0705-y.

T. Nguyen-Thoi, P. Phung-Van, H. Nguyen-Xuan, and C. Thai-Hoang. A cell-based smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner-Mindlin plates. International Journal for Numerical Methods in Engineering, 91, (7), (2012), pp. 705–741. doi:10.1002/nme.4289.

H. Nguyen-Xuan, G. R. Liu, C. Thai-Hoang, and T. Nguyen-Thoi. An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 199, (9), (2010), pp. 471–489. doi:10.1016/j.cma.2009.09.001.

H. Nguyen-Xuan, T. Rabczuk, N. Nguyen-Thanh, T. Nguyen-Thoi, and S. Bordas. A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates. Computational Mechanics, 46, (5), (2010), pp. 679–701. doi:10.1007/s00466-010-0509-x.

T. Nguyen-Thoi, P. Phung-Van, C. Thai-Hoang, and H. Nguyen-Xuan. A cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements for static and free vibration analyses of shell structures. International Journal of Mechanical Sciences, 74, (2013), pp. 32–45. doi:10.1016/j.ijmecsci.2013.04.005.

X. Cui, G. R. Liu, G. Y. Li, G. Y. Zhang, and G. Zheng. Analysis of plates and shells using an edge-based smoothed finite element method. Computational Mechanics, 45, (2), (2010), pp. 141–156. doi:10.1007/s00466-009-0429-9.

S. Nguyen-Hoang, P. Phung-Van, S. Natarajan, and H. G. Kim. A combined scheme of edge-based and node-based smoothed finite element methods for Reissner–Mindlin flat shells. Engineering with Computers, 32, (2), (2016), pp. 267–284. doi:10.1007/s00366-015-0416-z.

C. H. Thai, L. V. Tran, D. T. Tran, T. Nguyen-Thoi, and H. Nguyen-Xuan. Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method. Applied Mathematical Modelling, 36, (11), (2012), pp. 5657–5677. doi:10.1016/j.apm.2012.01.003.

P. Phung-Van, T. Nguyen-Thoi, L. V. Tran, and H. Nguyen-Xuan. A cell-based smoothed discrete shear gap method (CS-DSG3) based on the C0-type higher-order shear deformation theory for static and free vibration analyses of functionally graded plates. Computational Materials Science, 79, (2013), pp. 857–872. doi:10.1016/j.commatsci.2013.06.010.

L. Wu, P. Liu, C. Shi, Z. Zhang, T. Q. Bui, and D. Jiao. Edge-based smoothed extended finite element method for dynamic fracture analysis. Applied Mathematical Modelling, 40, (19), (2016), pp. 8564–8579. doi:10.1016/j.apm.2016.05.027.

P. Liu, T. Q. Bui, C. Zhang, T. T. Yu, G. R. Liu, and M. V. Golub. The singular edge-based smoothed finite element method for stationary dynamic crack problems in 2D elastic solids. Computer Methods in Applied Mechanics and Engineering, 233, (2012), pp. 68–80. doi:10.1016/j.cma.2012.04.008.

S. P. Timoshenko and S. Woinowsky-Krieger. Theory of plates and shells. McGraw-Hill, second edition, (1959).

M. Lyly, R. Stenberg, and T. Vihinen. A stable bilinear element for the Reissner-Mindlin plate model. Computer Methods in Applied Mechanics and Engineering, 110, (3-4), (1993), pp. 343–357. doi:10.1016/0045-7825(93)90214-i.

R. L. Taylor and F. Auricchio. Linked interpolation for Reissner-Mindlin plate elements: Part II–A simple triangle. International Journal for Numerical Methods in Engineering, 36, (18), (1993), pp. 3057–3066. doi:10.1002/nme.1620361803.

L. S. D. Morley. Skew plates and structures. Pergamon Press, Oxford, (1963).

R. Ayad, G. Dhatt, and J. L. Batoz. A new hybrid-mixed variational approach for Reissner-Mindlin plates. The MiSP model. International Journal for Numerical Methods in Engineering, 42, (7), (1998), pp. 1149–1179. doi:10.1002/(sici)1097-0207(19980815)42:7<1149::aid-nme391>3.0.co;2-2.

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*A cell-based smoothed three-node plate finite element with a bubble node for static analyses of both thin and thick plates*, Vietnam J. Mech.

**39**(2017) 229–243. DOI: https://doi.org/10.15625/0866-7136/8809.

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