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Research Papers

Improvements in Shear Locking and Spurious Zero Energy Modes Using Chebyshev Finite Element Method

[+] Author and Article Information
H. Dang-Trung

Division of Computational
Mathematics and Engineering,
Institute for Computational Science,
Ton Duc Thang University,
Ho Chi Minh City 700000, Vietnam;
Faculty of Civil Engineering,
Ton Duc Thang University,
Ho Chi Minh City 700000, Vietnam;
Department of Mathematics,
University of Bergen,
Bergen 5020, Norway
e-mails: dangtrunghau@tdt.edu.vn; dtrhau@gmail.com

Dane-Jong Yang

Department of Mechanical and
Computer-Aided Engineering,
Feng Chia University,
Seatwen,
Taichung 40724, Taiwan
e-mail: mgyu49@yahoo.com.tw

Y. C. Liu

Bachelor's Program in Precision System Design,
Feng Chia University,
Seatwen,
Taichung 40724, Taiwan
e-mail: yucliu@fcu.edu.tw

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received April 12, 2018; final manuscript received October 22, 2018; published online November 19, 2018. Assoc. Editor: John Michopoulos.

J. Comput. Inf. Sci. Eng 19(1), 011006 (Nov 19, 2018) (16 pages) Paper No: JCISE-18-1090; doi: 10.1115/1.4041829 History: Received April 12, 2018; Revised October 22, 2018

Abstract

In this paper, the authors present Chebyshev finite element (CFE) method for the analysis of Reissner–Mindlin (RM) plates and shells. Chebyshev polynomials are a sequence of orthogonal polynomials that are defined recursively. The values of the polynomials belong to the interval $[−1,1]$ and vanish at the Gauss points (GPs). Therefore, high-order shape functions, which satisfy the interpolation condition at the points, can be performed with Chebyshev polynomials. Full gauss quadrature rule was used for stiffness matrix, mass matrix and load vector calculations. Static and free vibration analyses of thick and thin plates and shells of different shapes subjected to different boundary conditions were conducted. Both regular and irregular meshes were considered. The results showed that by increasing the order of the shape functions, CFE automatically overcomes shear locking without the formation of spurious zero energy modes. Moreover, the results of CFE are in close agreement with the exact solutions even for coarse and irregular meshes.

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Figures

Fig. 1

Positive directions of the displacement w and two rotations βx and βy

Fig. 2

Third-order shape functions based on Chebyshev polynomials in one-dimensional space

Fig. 3

Runge's phenomenon: dash curve—LIP and solid curve—CIP

Fig. 4

(a) Clamped square plate, (b) 2 × 2 elements, and (c) 8 × 8 elements

Fig. 5

Convergence curves of the CFE for a fully clamped square plate

Fig. 6

Convergence curves of the CFE for a fully simply supported square plate

Fig. 7

Performance of CFE with varying length-to-thickness values: (a) clamped plate and (b) simply supported plate

Fig. 8

Comparison of convergences of CFE with MITC4 and MITC9 elements for thin square plate (t/L = 0.0001): (a) clamped plate and (b) simply supported plate

Fig. 9

Mesh distortion of a square plate: (a) s = 0, (b) s = 0.3, and (c) s = 0.5

Fig. 10

(a) A clamped rhombic plate and (b) 8 × 8 elements

Fig. 11

(a) A circular plate, (b) 16 elements, and (c) 256 elements

Fig. 12

Comparison of the first six resonance frequencies of a square thick plate: (a) clamped plate and (b) simply supported plate

Fig. 13

Comparison of the first six resonance frequencies of a square thin plate: (a) clamped plate and (b) simply supported plate

Fig. 14

The first six mode shapes of a simply supported square thick plate (t/L=0.1)

Fig. 15

Comparisons of the first six resonance frequencies of a cantilevered rhombic plate (α=60 deg): (a) thick plate and (b) thin plate

Fig. 16

The first six mode shapes of thin, cantilevered, rhombic plate (t/L=0.001, α=60 deg)

Fig. 17

Comparison of first six frequencies of a clamped circular plate: (a) thick plate and (b) thin plate

Fig. 18

The first six mode shapes of clamped circular plate (t/R = 0.02)

Fig. 19

Cylindrical shell problem: (a) cylindrical shell model, (b) deformation of cylindrical shell by CFE, and (c) deformation of cylindrical shell by ANSYS

Fig. 20

Comparison the convergence of center displacement of cylindrical shell under concentrated loads

Fig. 21

The various mode shapes of cantilever cylindrical shell

Fig. 22

Spherical shell problem: (a) spherical shell model and (b) deformation of spherical shell under concentrated load

Fig. 23

Comparison of the convergence of center displacement of spherical shell under concentrated load

Fig. 24

The first six mode shapes of spherical shell

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