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

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
H. Dang-Trung ◽  
Dane-Jong Yang ◽  
Y. C. Liu
Keyword(s):  
Finite Element ◽  
Chebyshev Polynomials ◽  
Mass Matrix ◽  
Thin Plates ◽  
Shear Locking ◽  
Shape Functions ◽  
Zero Energy ◽  
Interpolation Condition ◽  
Plates And Shells ◽  
Irregular Meshes

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.

Efficient Analysis of 3-D Plates via Algebraic Reduction

2009 ◽  
Author(s):  
Vikalp Mishra ◽  
Krishnan Suresh
Keyword(s):  
Finite Element ◽  
Dimensional Space ◽  
Reduction Process ◽  
Thin Plates ◽  
Computationally Efficient ◽  
Design Environment ◽  
Element Analysis ◽  
Plates And Shells ◽  
Algebraic Reduction ◽  
Lower Dimensional Space

3-D finite element analysis (3-D FEA) is not generally recommended for analyzing thin structures such as plates and shells. Instead, a variety of highly efficient and specialized 2-D numerical methods have been developed for analyzing such structures. However, 2-D methods pose serious automation challenges in today’s 3-D design environment. In this paper, we propose an efficient yet easily automatable 3-D algebraic reduction method for analyzing thin plates. The proposed method exploits standard off-the-shelf finite element packages, and it achieves high computational efficiency through an algebraic reduction process. In the reduction process, a 3-D plate bending stiffness matrix is constructed from a 3-D mesh, and then projected onto a lower-dimensional space by appealing to standard 2-D plate-theories. Algebraic reduction offers the best of both worlds in that it is computationally efficient, and yet easy to automate. The proposed methodology is substantiated through numerical experiments.

Finite Element Modeling of Dynamic Behavior of Some Basic Structural Members

1992 ◽  
Vol 114 (1) ◽  
pp. 3-9 ◽  
Author(s):  
R. C. Engels
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Modeling ◽  
Mass Matrix ◽  
Matrix Approach ◽  
Structural Members ◽  
Shape Functions ◽  
Generalized Coordinates ◽  
Stiffness Matrices ◽  
Consistent Mass Matrix

A method is described to model the dynamics of finite elements. The assumed modes method is used to show how static shape functions approximate the element mass distribution. The deterioration of the modal content of a model can be linked to the neglect of interface restrained assumed modes. Restoration of a few of these modes leads to higher accuracy with fewer generalized coordinates compared to the standard consistent mass matrix approach. Also, no need exists for subdivision of basic elements such as rods and beams. The mass and stiffness matrices for several basic elements are derived and used in demonstration problems.

A Universal Finite Element of Thick and Thin Plate without Shear Locking

2011 ◽  
Vol 52-54 ◽  
pp. 1353-1357
Author(s):  
Shu Qiang Yu ◽  
Ming Zhang ◽  
Lu Lu Fan
Keyword(s):  
Finite Element ◽  
Thin Plate ◽  
Plate Thickness ◽  
High Accuracy ◽  
Thin Plates ◽  
Deep Beam ◽  
Computational Results ◽  
Shear Locking ◽  
Plate Element ◽  
Universal Element

In order to prevent shear locking, a method using theory of deep beam is proposed. A universal finite element for thick and thin plates is constructed. When the plate thickness approaches to the limit of thin plate, the universal element degenerates to the thin plate element automatically. As a results, the shear locking phenomenon will not appear. The computational results indicate that the current element has high-accuracy and good usefulness.

Analysis of Multilayered/Sandwich Shells Using a New Discrete Finite Element Model

2013 ◽  
Vol 682 ◽  
pp. 185-190
Author(s):  
S. Sakami ◽  
H. Sabhi ◽  
R. Ayad
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Dynamic Analysis ◽  
Element Model ◽  
Thin Plates ◽  
Shear Locking ◽  
Patch Tests ◽  
Static And Dynamic Analysis ◽  
Quadratic Interpolation ◽  
Discrete Finite Element

The model DDM (Discrete Displacement Mindlin), leads to a finite element which is geometrically simple: 4-node quadrilateral with 5 doffs per node for a shell and efficient. The mid-side rotational nodes, derived from a quadratic interpolation of normal rotations, are eliminated using a combination of local discrete kinematic and constitutive Mindlin hypotheses. The derived 4-node element is free of shear locking and passes all patch tests for thick and thin plates in arbitrary mesh. The applications concern the static and dynamic analysis of sandwich and multilayered shells.

Frequency Response of a Three-Node Finite Element for Thick and Thin Plates

2002 ◽  
Vol 8 (8) ◽  
pp. 1123-1153 ◽  
Author(s):  
Humayun R. H. Kabir ◽  
Abdullateef M. Al-Khaleefi
Keyword(s):  
Finite Element ◽  
Frequency Response ◽  
Correction Term ◽  
Thin Plates ◽  
Mode Shapes ◽  
Integration Scheme ◽  
Element Formulation ◽  
Transverse Shear Strain ◽  
Shear Locking ◽  
Triangular Finite Element

A shear-locking free isoparametric three-node triangular finite element is presented to study the frequency response of moderately thick and thin plates. Reissner/Mindlin theory that incorporates shear deformation effects is included into the element formulation. A shear correction term is introduced in transverse shear strain components to avoid the shear-locking phenomenon. The element is developed with a full integration scheme, hence, the element remains kinematically stable. Natural frequencies and mode shapes are obtained and compared with the available analytical and finite element solutions.

Conforming shear-locking-free four-node rectangular finite element of moderately thick plate

2016 ◽  
Vol 25 (5-6) ◽  
pp. 141-152
Author(s):  
Ivo Senjanović ◽  
Marko Tomić ◽  
Smiljko Rudan ◽  
Neven Hadžić
Keyword(s):  
Finite Element ◽  
Thick Plate ◽  
Plate Theory ◽  
Shear Stiffness ◽  
Mindlin Plate ◽  
Shear Locking ◽  
Shape Functions ◽  
Beam Shape ◽  
Stiffness Matrices ◽  
Moderately Thick Plate

AbstractAn outline of the modified Mindlin plate theory, which deals with bending deflection as a single variable, is presented. Shear deflection and cross-section rotation angles are functions of bending deflection. A new four-node rectangular finite element of moderately thick plate is formulated by utilizing the modified Mindlin theory. Shape functions of total (bending+shear) deflections are defined as a product of the Timshenko beam shape functions in the plate longitudinal and transversal direction. The bending and shear stiffness matrices, and translational and rotary mass matrices are specified. In this way conforming and shear-locking-free finite element is obtained. Numerical examples of plate vibration analysis, performed for various combinations of boundary conditions, show high level of accuracy and monotonic convergence of natural frequencies to analytical values. The new finite element is superior to some sophisticated finite elements incorporated in commercial software.

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