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 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.
Issue Section:
Research Papers
References
1.
Abassian, F., Haswell, D. J., and Knowles, N. C., 1987, “
Free Vibration Benchmarks
,” National Agency for Finite Element Methods and Standards, Glasgow, UK.2.
Al-Bermani
, F. G. A.
, and Liew
, K. M.
, 1996
, “Natural Frequencies of Thick Arbitrary Quadrilateral Plates Using the pb–2 Ritz Method
,” J. Sound Vib.
, 196
(4
), pp. 371
–385
.3.
Ayad
, R.
, Dhatt
, G.
, and Batoz
, J. L.
, 1998
, “A New Hybrid-Mixed Variational Approach for Reissner–Mindlin Plates. The MiSP Model
,” Int. J. Numer. Methods Eng.
, 42
(7
), pp. 1149
–1179
.4.
Irie
, T.
, Yamada
, G.
, and Aomura
, S.
, 1980
, “Natural Frequencies of Mindlin Circular Plates
,” ASME J. Appl. Mech.
, 47
(3
), pp. 652
–655
.5.
Taylor
, R. L.
, and Auricchio
, F.
, 1993
, “Linked Interpolation for Reissner-Mindlin Plate Elements—Part II: A Simple Triangle
,” Int. J. Numer. Methods Eng.
, 36
(18
), pp. 3057
–3066
.6.
Timoshenko
, S.
, and Woinowsky-Krieger
, S.
, 1959
, Theory of Plates and Shells
(Engineering Societies Monographs
), McGraw-Hill
, New York.7.
Szilard
, R.
, 2004
, Theories and Applications of Plate Analysis: Classical Numerical and Engineering Methods
, Wiley
, Hoboken, NJ.8.
Clough
, R. W.
, 1960
, The Finite Element Method in Plane Stress Analysis
, American Society of Civil Engineers
, Reston, VA.9.
Love
, A. E. H.
, 1888
, “The Small Free Vibrations and Deformation of a Thin Elastic Shell
,” Philos. Trans. R. Soc. London
, 179
, pp. 491–546.https://www.jstor.org/stable/9052710.
Mindlin
, R. D.
, 1951
, “Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic Elastic Pates
,” ASME J. Appl. Mech.
, 18
, pp. 31
–38
.11.
Reissner
, E.
, 1945
, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates
,” ASME J. Appl. Mech.
, 12
, pp. A68
–A77
.12.
Zienkiewicz
, O. C.
, and Taylor
, R. L.
, 2000
, The Finite Element Method: Solid Mechanics
, Butterworth-Heinemann
, Oxford, UK.13.
Bathe
, K.-J.
, and Dvorkin
, E. N.
, 1986
, “A Formulation of General Shell Elements—The Use of Mixed Interpolation of Tensorial Components
,” Int. J. Numer. Methods Eng.
, 22
(3
), pp. 697
–722
.14.
Ko
, Y.
, Lee
, P.-S.
, and Bathe
, K.-J.
, 2016
, “The MITC4+ Shell Element and Its Performance
,” Comput. Struct.
, 169
, pp. 57
–68
.15.
Bathe
, K.-J.
, Lee
, P.-S.
, and Hiller
, J.-F.
, 2003
, “Towards Improving the MITC9 Shell Element
,” Comput. Struct.
, 81
(8–11
), pp. 477
–489
.16.
Bucalem
, M. L.
, and Bathe
, K.-J.
, 1993
, “Higher-Order MITC General Shell Elements
,” Int. J. Numer. Methods Eng.
, 36
(21
), pp. 3729
–3754
.17.
Beirão da Veiga
, L.
, Chapelle
, D.
, and Paris Suarez
, I.
, 2007
, “Towards Improving the MITC6 Triangular Shell Element
,” Comput. Struct.
, 85
(21–22
), pp. 1589
–1610
.18.
Bathe
, K.-J.
, Brezzi
, F.
, and Cho
, S. W.
, 1989
, “The MITC7 and MITC9 Plate Bending Elements
,” Comput. Struct.
, 32
(3–4
), pp. 797
–814
.19.
Bletzinger
, K.-U.
, Bischoff
, M.
, and Ramm
, E.
, 2000
, “A Unified Approach for Shear-Locking-Free Triangular and Rectangular Shell Finite Elements
,” Comput. Struct.
, 75
(3
), pp. 321
–334
.20.
Tessler
, A.
, and Hughes
, T. J. R.
, 1985
, “A Three-Node Mindlin Plate Element With Improved Transverse Shear
,” Comput. Methods Appl. Mech. Eng.
, 50
(1
), pp. 71
–101
.21.
Tessler
, A.
, and Hughes
, T. J. R.
, 1983
, “An Improved Treatment of Transverse Shear in the Mindlin-Type Four-Node Quadrilateral Element
,” Comput. Methods Appl. Mech. Eng.
, 39
(3
), pp. 311
–335
.22.
Zrahia
, U.
, and Bar-Yoseph
, P.
, 1995
, “Plate Spectral Elements Based Upon Reissner–Mindlin Theory
,” Int. J. Numer. Methods Eng.
, 38
(8
), pp. 1341
–1360
.23.
Brito
, K. D.
, and Sprague
, M. A.
, 2012
, “Reissner–Mindlin Legendre Spectral Finite Elements With Mixed Reduced Quadrature
,” Finite Elem. Anal. Des.
, 58
, pp. 74
–83
.24.
Sprague
, M. A.
, and Purkayastha
, A.
, 2015
, “Legendre Spectral Finite Elements for Reissner–Mindlin Composite Plates
,” Finite Elem. Anal. Des.
, 105
, pp. 33
–43
.25.
Tchébychev, M. P., 1954, “
Théorie des mécanismes connus sous le nom de parallélogrammes
,” Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg, 7
, pp. 539–586.26.
Patera
, A. T.
, 1984
, “A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion
,” J. Comput. Phys.
, 54
(3
), pp. 468
–488
.27.
Canuto
, C.
, Quarteroni
, A.
, Hussaini
, M. Y.
, and Zang
, T. A.
, 2012
, “Spectral Methods in Fluid Dynamics
,” Springer, New York.28.
Deville
, M. O.
, 1990
, “Chebyshev Collocation Solutions of Flow Problems
,” Comput. Methods Appl. Mech. Eng.
, 80
(1–3
), pp. 27
–37
.29.
Lee-Wing
, H.
, and Patera
, A. T.
, 1990
, “A Legendre Spectral Element Method for Simulation of Unsteady Incompressible Viscous Free-Surface Flows
,” Comput. Methods Appl. Mech. Eng.
, 80
(1–3
), pp. 355
–366
.30.
Komatitsch
, D.
, Vilotte
, J.-P.
, Vai
, R.
, Castillo-Covarrubias
, J. M.
, and Sánchez-Sesma
, F. J.
, 1999
, “The Spectral Element Method for Elastic Wave Equations—Application to 2-D and 3-D Seismic Problems
,” Int. J. Numer. Methods Eng.
, 45
(9
), pp. 1139
–1164
.31.
Seriani
, G.
, 1998
, “3-D Large-Scale Wave Propagation Modeling by Spectral Element Method on Cray T3E Multiprocessor
,” Comput. Methods Appl. Mech. Eng.
, 164
(1–2
), pp. 235
–247
.32.
Seriani
, G.
, 1997
, “A Parallel Spectral Element Method for Acoustic Wave Modeling
,” J. Comput. Acoust.
, 5
(1
), pp. 53
–69
.33.
Dauksher
, W.
, and Emery
, A. F.
, 1997
, “Accuracy in Modeling the Acoustic Wave Equation With Chebyshev Spectral Finite Elements
,” Finite Elem. Anal. Des.
, 26
(2
), pp. 115
–128
.34.
Seriani
, G.
, and Priolo
, E.
, 1994
, “Spectral Element Method for Acoustic Wave Simulation in Heterogeneous Media
,” Finite Elem. Anal. Des.
, 16
(3–4
), pp. 337
–348
.35.
Seriani
, G.
, Priolo
, E.
, Carcione
, J.
, and Padovani
, E.
, 1992
, “High-Order Spectral Element Method for Elastic Wave Modeling
,” SEG Technical Program Expanded Abstracts 1992
, Society of Exploration Geophysicists, Tulsa, OK, pp. 1285
–1288
.36.
Nguyen
, H. D.
, Paik
, S.
, and Douglass
, R. W.
, 1997
, “A Legendre-Spectral Element Method for Flow and Heat Transfer About an Accelerating Droplet
,” J. Sci. Comput.
, 12
, pp. 75
–97
.37.
Spall
, R.
, 1995
, “Spectral Collocation Methods for One-Dimensional Phase-Change Problems
,” Int. J. Heat Mass Transfer
, 38
(15
), pp. 2743
–2748
.38.
Dauksher
, W.
, and Emery
, A. F.
, 2000
, “The Solution of Elastostatic and Elastodynamic Problems With Chebyshev Spectral Finite Elements
,” Comput. Methods Appl. Mech. Eng.
, 188
(1–3
), pp. 217
–233
.39.
Lee
, U.
, 2009
, Spectral Element Method in Structural Dynamics
, Wiley
, Singapore.40.
Liu
, G. R.
, and Quek
, S. S.
, 2003
, Finite Element Method: A Practical Course
, Butterworth-Heinemann
, Oxford, UK
, pp. 1
–11
.41.
Nguyen-Thoi
, T.
, Phung-Van
, P.
, Thai-Hoang
, C.
, and Nguyen-Xuan
, H.
, 2013
, “A Cell-Based Smoothed Discrete Shear Gap Method (CS-DSG3) Using Triangular Elements for Static and Free Vibration Analyses of Shell Structures
,” Int. J. Mech. Sci.
, 74
, pp. 32
–45
.42.
Fornberg
, B.
, and Zuev
, J.
, 2007
, “The Runge Phenomenon and Spatially Variable Shape Parameters in RBF Interpolation
,” Comput. Math. Appl.
, 54
(3), pp. 379–398.43.
Nguyen-Thoi
, M. H.
, Le-Anh
, L.
, Ho-Huu
, V.
, Dang-Trung
, H.
, and Nguyen-Thoi
, T.
, 2015
, “An Extended Cell-Based Smoothed Discrete Shear Gap Method (XCS-FEM-DSG3) for Free Vibration Analysis of Cracked Reissner-Mindlin Shells
,” Front. Struct. Civ. Eng.
, 9
(4
), pp. 341
–358
.44.
Nguyen-Thoi
, T.
, Phung-Van
, P.
, Nguyen-Xuan
, H.
, and Thai-Hoang
, C.
, 2012
, “A Cell-Based Smoothed Discrete Shear Gap Method Using Triangular Elements for Static and Free Vibration Analyses of Reissner–Mindlin Plates
,” Int. J. Numer. Methods Eng.
, 91
(7
), pp. 705
–741
.45.
Hiller
, J.-F.
, and Bathe
, K.-J.
, 2003
, “Measuring Convergence of Mixed Finite Element Discretizations: An Application to Shell Structures
,” Comput. Struct.
, 81
(8–11
), pp. 639
–654
.46.
Bathe
, K.-J.
, 2014
, Finite Element Procedures
. Prentice Hall, Upper Saddle River, NJ.47.
Chapelle
, D.
, and Bathe
, K.-J.
, 2011
, The Finite Element Analysis of Shells—Fundamentals
, Springer-Verlag, Berlin.48.
Nguyen-Xuan
, H.
, Rabczuk
, T.
, Bordas
, S.
, and Debongnie
, J. F.
, 2008
, “A Smoothed Finite Element Method for Plate Analysis
,” Comput. Methods Appl. Mech. Eng.
, 197
(13–16
), pp. 1184
–1203
.49.
Rao
, H. V. S. G.
, and Chaudhary
, V. K.
, 1988
, “Analysis of Skew and Triangular Plates in Bending
,” Comput. Struct.
, 28
(2
), pp. 223
–235
.50.
Morley
, L. S. D.
, 1962
, “Bending of a Simply Supported Rhombic Plate Under Uniform Normal Loading
,” Q. J. Mech. Appl. Math.
, 15
(4
), pp. 413
–426
.51.
Häggblad
, B.
, and Bathe
, K.-J.
, 1990
, “Specifications of Boundary Conditions for Reissner/Mindlin Plate Bending Finite Elements
,” Int. J. Numer. Methods Eng.
, 30
(5
), pp. 981
–1011
.52.
Leissa
, A. W.
, 1969
, “Vibration of Plates
,” Ohio State University, Columbus, OH, p. 362
.53.
Lee
, S. J.
, 2004
, “Free Vibration Analysis of Plates by Using a Four-Node Finite Element Formulated With Assumed Natural Transverse Shear Strain
,” J. Sound Vib.
, 278
(3
), pp. 657
–684
.54.
Shi
, X.
, Shi
, D.
, Li
, W. L.
, and Wang
, Q.
, 2014
, “A Unified Method for Free Vibration Analysis of Circular, Annular and Sector Plates With Arbitrary Boundary Conditions
,” J. Vib. Control
, 22
(2
), pp. 442
–456
.55.
Shojaee
, S.
, Izadpanah
, E.
, Valizadeh
, N.
, and Kiendl
, J.
, 2012
, “Free Vibration Analysis of Thin Plates by Using a NURBS-Based Isogeometric Approach
,” Finite Elem. Anal. Des.
, 61
, pp. 23
–34
.56.
Krysl
, P.
, and Belytschko
, T.
, 1996
, “Analysis of Thin Shells by the Element-Free Galerkin Method
,” Int. J. Solids Struct.
, 33
(20–22
), pp. 3057
–3080
.57.
Noguchi
, H.
, Kawashima
, T.
, and Miyamura
, T.
, 2000
, “Element Free Analyses of Shell and Spatial Structures
,” Int. J. Numer. Methods Eng.
, 47
(6
), pp. 1215
–1240
.58.
Liew
, K. M.
, Peng
, L. X.
, and Ng
, T. Y.
, 2002
, “Three-Dimensional Vibration Analysis of Spherical Shell Panels Subjected to Different Boundary Conditions
,” Int. J. Mech. Sci.
, 44
(10
), pp. 2103
–2117
.Copyright © 2019 by ASME
You do not currently have access to this content.
