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

## DOI:

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.

### Downloads

### Metrics

## References

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.

## Downloads

## Published

## How to Cite

*Vietnam Journal of Mechanics*,

*39*(3), 229–243. https://doi.org/10.15625/0866-7136/8809

## Issue

## Section

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.