Nonlinear Vibration of Thin Elastic Plates, Part 1: Generalized Incremental Hamilton’s Principle and Element Formulation

1984 ◽  
Vol 51 (4) ◽  
pp. 837-844 ◽  
Author(s):  
S. L. Lau ◽  
Y. K. Cheung ◽  
S. Y. Wu
Keyword(s):  
Nonlinear Problems ◽  
Free Vibrations ◽  
Element Formulation ◽  
Hamilton’S Principle ◽  
Elastic Plates ◽  
Hamilton's Principle ◽  
Nonlinear Bending ◽  
Nodal Displacements ◽  
Kirchhoff Theory ◽  
Thin Elastic Plates

The finite element method has been widely used for analyzing nonlinear problems, but it is surprising that so far only a few papers have been devoted to nonlinear periodic structural vibrations. In Part 1 of this paper, a generalized incremental Hamilton’s principle for nonlinear periodic vibrations of thin elastic plates is presented. This principle is particularly suitable for the formulation of finite elements and finite strips in geometrically nonlinear plate problems due to the fact that the nonlinear parts of inplane stress resultants are functions subject to variation and that the Kirchhoff assumption is included as part of its Euler equations. Following a general formulation method given in this paper, a simple triangular incremental modified Discrete Kirchhoff Theory (DKT) plate element with 15 stretching and bending nodal displacements is derived. The accuracy of this element is demonstrated via some typical examples of nonlinear bending and frequency response of free vibrations. Comparisons with previous results are also made. In Part 2 of this paper, this incremental element is applied to the computation of complicated frequency responses of plates with existence of internal resonance and very interesting seminumerical results are obtained.

scholarly journals A Corotational Formulation Based on Hamilton’s Principle for Geometrically Nonlinear Thin and Thick Planar Beams and Frames

2018 ◽  
Vol 2018 ◽  
pp. 1-22 ◽  
Author(s):  
Hesham A. Elkaranshawy ◽  
Ahmed A. H. Elerian ◽  
Walied I. Hussien
Keyword(s):  
Nonlinear Behavior ◽  
Body Motion ◽  
Global Coordinate System ◽  
Element Formulation ◽  
Geometrically Nonlinear ◽  
Hamilton’S Principle ◽  
Equilibrium Equations ◽  
Hamilton's Principle ◽  
Time Saving ◽  
Corotational Formulation

A corotational finite element formulation for two-dimensional beam elements with geometrically nonlinear behavior is presented. The formulation separates the rigid body motion from the pure deformation which is always small relative to the corotational element frame. The stiffness matrices and the mass matrices are evaluated using both Euler-Bernoulli and Timoshenko beam models to reveal the shear effect in thin and thick beams and frames. The nonlinear equilibrium equations are developed using Hamilton’s principle and are defined in the global coordinate system. A MATLAB code is developed for the numerical solution. In static analysis, the code employed an iterative method based on the full Newton-Raphson method without incremental loading, while, in dynamic analysis, the Newmark direct integration implicit method is also utilized. Several examples of flexible beams and frames with large displacements are presented. Not only is the method simple and time-saving, but it is also highly effective and highly accurate.

scholarly journals Self-dual singularity through lasing and antilasing in thin elastic plates

2021 ◽  
Vol 103 (13) ◽  
Author(s):  
M. Farhat ◽  
P.-Y. Chen ◽  
S. Guenneau ◽  
Y. Wu
Keyword(s):  
Elastic Plates ◽  
Thin Elastic Plates

scholarly journals Spacetime modulation in floating thin elastic plates

2021 ◽  
Vol 104 (1) ◽  
Author(s):  
Mohamed Farhat ◽  
Sebastien Guenneau ◽  
Pai-Yen Chen ◽  
Ying Wu
Keyword(s):  
Elastic Plates ◽  
Thin Elastic Plates

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky
Keyword(s):  
Mathematical Theory ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Theory Of Elasticity ◽  
Elastic Plates ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Complex Boundary ◽  
Plates And Shells ◽  
Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

Sign in / Sign up

Export Citation Format

Share Document