A Numerical Solution to the General Large Deflection Plate Equations

1965 ◽  
Author(s):  
Parker Holmes Petit
Keyword(s):  
Numerical Solution ◽  
Large Deflection ◽  
Deflection Plate ◽  
Plate Equations

Comparison of the Two Formulations of w-u-v and w-F in Nonlinear Plate Analysis

2002 ◽  
Vol 69 (4) ◽  
pp. 547-552 ◽  
Author(s):  
J. Lee
Keyword(s):  
Large Deflection ◽  
Plate Theory ◽  
Airy Function ◽  
Transverse Displacement ◽  
Simply Supported ◽  
Deflection Plate ◽  
Plate Equations ◽  
Plate Analysis ◽  
Plane Displacement ◽  
Hamiltonian Property

In a moderately large deflection plate theory of von Karman and Chu-Herrmann, one may consider thin-plate equations of either the transverse and in-plane displacements, w-u-v formulation, or the transverse displacement and Airy function, w-F formulation. Under the Galerkin procedure, we examine if the modal equations of two plate formulations preserve the Hamiltonian property which demands energy conservation in the conservative limit of no damping and forcing. In the w-F formulation, we have shown that modal equations are Hamiltonian for the first four symmetric modes of a simply-supported plate. In contrast, the corresponding modal equations of w-u-v formulation do not exhibit the Hamiltonian property when a finite number of sine terms are included in the in-plane displacement expansions.

Dynamic Elastic/Viscoplastic Response of Hull Plating Subjected to Hydrodynamic Wave Impact

1997 ◽  
Vol 41 (02) ◽  
pp. 130-146
Author(s):  
P. A. Caridis ◽  
M. Stefanou
Keyword(s):  
Large Deflection ◽  
Pressure Pulse ◽  
Material Behavior ◽  
Maximum Pressure ◽  
Wave Impact ◽  
Multiple Wave ◽  
Deflection Plate ◽  
Model Collision ◽  
Plate Equations ◽  
Hydrodynamic Wave

A numerical simulation of the response of a flat plate subjected to hydrodynamic wave impact is presented. The formulation is based on a time domain solution of the large deflection plate equations using a finite difference mesh. The material behavior is elastic/viscoplastic, following the classical Perzyna model, and the procedure is validated against a series of model collision experiments. The correlation results were found to be good and the procedure was subsequently used to predict the response of plating subjected to wave impact loads, characterized by a steep rise to a maximum pressure followed by an exponential decay. A comparison with an analytical model was conducted and following this a parametric study of the characteristics of the pressure pulse was carried out. The paper concludes with a study of the effect of multiple wave impacts. It was found that, even though individual impacts may not cause rupture, a small number of these may lead to rupture of the plating, provided the maximum pressure is large enough.

A Model of a Trapped Particle Under a Plate Adhering to a Rigid Surface

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Joel R. Parent ◽  
George G. Adams
Keyword(s):  
Numerical Solution ◽  
Large Deflection ◽  
Nonlinear Differential Equations ◽  
Plate Thickness ◽  
Work Of Adhesion ◽  
Contact Radius ◽  
Approximate Analytic Solution ◽  
Total Potential ◽  
Curve Fit ◽  
Deflection Theory

Micro and nanomechanics are growing fields in the semiconductor and related industries. Consequently obstacles, such as particles trapped between layers, are becoming more important and warrant further attention. In this paper a numerical solution to the von Kármán equations for moderately large deflection is used to model a plate deformed due to a trapped particle lying between it and a rigid substrate. Due to the small scales involved, the effect of adhesion is included. The recently developed moment-discontinuity method is used to relate the work of adhesion to the contact radius without the explicit need to calculate the total potential energy. Three different boundary conditions are considered—the full clamp, the partial clamp, and the compliant clamp. Curve-fit equations are found for the numerical solution to the nondimensional coupled nonlinear differential equations for moderately large deflection of an axisymmetric plate. These results are found to match the small deflection theory when the deflection is less than the plate thickness. When the maximum deflection is much greater than the plate thickness, these results represent the membrane theory for which an approximate analytic solution exists.

On the Large Axisymmetrical Deflection of Thin Plates Made of the Mooney-Rivlin Material

1974 ◽  
Vol 41 (3) ◽  
pp. 725-730 ◽  
Author(s):  
H. Abe´ ◽  
M. Utsui
Keyword(s):  
Large Deflection ◽  
Lateral Pressure ◽  
Three Dimensional ◽  
Thin Plates ◽  
Axially Symmetric ◽  
Dimensional Theory ◽  
Plate Equations ◽  
Basic Equations ◽  
Deflection Theory ◽  
Large Deflection Theory

A large deflection theory of axially symmetric and thin plates made of the Mooney-Rivlin material is developed by making a systematic and consistent approximation from the exact three-dimensional theory. The problem of a circular plate made of the neo-Hookean material subjected to uniform lateral pressure is investigated with the use of the basic equations just derived, and the results are compared with the solutions based on the von Karman plate equations.

The double mode model of the chaotic motion for a large deflection plate

1999 ◽  
Vol 20 (4) ◽  
pp. 360-364 ◽  
Author(s):  
Shu Xuefeng ◽  
Han Qiang ◽  
Yang Guitong
Keyword(s):  
Chaotic Motion ◽  
Large Deflection ◽  
Deflection Plate

scholarly journals Large Deflection of Prismatic Cantilever Beam Exposed to Combination of End Inclined Force and Tip Moment

2017 ◽  
Vol 12 (1) ◽  
pp. 98 ◽  
Author(s):  
Ibrahim Abu-Alshaikh ◽  
Hashem S. Alkhaldi ◽  
Nabil Beithou
Keyword(s):  
Numerical Solution ◽  
Cantilever Beam ◽  
Large Deflection ◽  
Elliptic Integral ◽  
Geometrical Nonlinearity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Integration Constant ◽  
Computational Time ◽  
Special Cases

The large deflection of a prismatic Euler-Bernoulli cantilever beam under a combination of end-concentrated coplanar inclined force and tip-concentrated moment is investigated. The angle of inclination of the applied force with respect to the horizontal axis remains unchanged during deformation. The cantilever beam is assumed to be naturally straight, slender, inextensible and elastic. The large deflection of the cantilever beam induces geometrical nonlinearity; hence, the nonlinear theory of bending and the exact expression of curvature are used. Based on an elliptic integral formulation, an accurate numerical solution is obtained in terms of an integration constant that should satisfy the boundary conditions associated with the cantilever beam. For some special cases this integration constant is exactly found, which leads to closed form solution. The numerical solution obtained is quite simple, accurate and involves less computational time compared with other techniques available in literature. The details of elastica and its corresponding orientation curves are presented and analyzed for extremely large load combinations. A comparative study with pre-obtained results has been made to verify the accuracy of the presented solution; an excellent agreement has been obtained.

Large Deflection of Plates Under Hydrostatic Pressure with Special Reference to Liquid Containers

1972 ◽  
Vol 16 (04) ◽  
pp. 261-270
Author(s):  
B. Aalami
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Special Reference ◽  
Large Deflection ◽  
Finite Difference Approximations ◽  
Isotropic Plates ◽  
Plate Equations ◽  
Bending Stresses ◽  
Order Of Magnitude ◽  
Nonlinear Plate Equations

A large-deflection stress analysis is made for square plates under hydrostatic pressure with several flexural and membrane boundary conditions, and with special reference to conditions related to flat-plate components in liquid containers and partitions. The analysis is based on von Karman's nonlinear plate equations for elastic isotropic plates using graded-mesh finite-difference approximations together with an iterative procedure. The influence on plate behavior of membrane and flexural boundary conditions is discussed. It is concluded that in thin-plated containers membrane stresses of the same order of magnitude as bending stresses develop. Solutions are offered nondimensionally in a tabular form for a number of more frequent membrane and bending boundary conditions suitable for design purposes. The application of the solutions is illustrated through numerical examples.

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