scholarly journals Stable and unstable bifurcation in the von Kármán problem for a circular plate

2005 ◽  
Vol 2005 (8) ◽  
pp. 889-899 ◽  
Author(s):  
Andrei Borisovich ◽  
Joanna Janczewska
Keyword(s):  
Circular Plate ◽  
Elastic Plate ◽  
Analytical Methods ◽  
Compressive Force ◽  
Elastic Base ◽  
Solution Set ◽  
Simply Supported ◽  
Bifurcation Points ◽  
Von Karman ◽  
Von Kármán

In this work, we study bifurcation in the von Kármán equations for a thin circular elastic plate which lies on the elastic base and is simply supported and subjected to a compressive force along the boundary. Applying analytical methods, we prove the existence of stable and unstable simple bifurcation points in the solution set of these equations.

ON A NONLOCAL FUNCTIONAL ARISING IN THE STUDY OF THIN-FILM BLISTERING

2010 ◽  
Vol 08 (02) ◽  
pp. 109-123
Author(s):  
N. ANSINI ◽  
V. VALENTE
Keyword(s):  
Thin Film ◽  
Asymptotic Analysis ◽  
Circular Plate ◽  
One Dimensional ◽  
Von Karman ◽  
Nonlocal Functional ◽  
Γ Convergence ◽  
Von Kármán

The energy of a Von Kármán circular plate is described by a nonlocal nonconvex one-dimensional functional depending on the thickness ε. Here we perform the asymptotic analysis via Γ-convergence as the parameter ε goes to zero.

On Von Kármán Equations and the Buckling of a Thin Circular Elastic Plate

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Joanna Janczewska ◽  
Anita Zgorzelska
Keyword(s):  
Elastic Plate ◽  
Compressive Load ◽  
Load Parameter ◽  
Elastic Plates ◽  
Plate Structures ◽  
Von Karman ◽  
Fredholm Map ◽  
Von Kármán Equations ◽  
Von Kármán ◽  
Nonlinear Deformations

AbstractWe shall be concerned with the buckling of a thin circular elastic plate simply supported along a boundary, subjected to a radial compressive load uniformly distributed along its boundary. One of the main engineering concerns is to reduce deformations of plate structures. It is well known that von Kármán equations provide an established model that describes nonlinear deformations of elastic plates. Our approach to study plate deformations is based on bifurcation theory. We will find critical values of the compressive load parameter by reducing von Kármán equations to an operator equation in Hölder spaces with a nonlinear Fredholm map of index zero. We will prove a sufficient condition for bifurcation by the use of a gradient version of the Crandall-Rabinowitz theorem due to A.Yu. Borisovich and basic notions of representation theory. Moreover, applying the key function method by Yu.I. Sapronov we will investigate the shape of bifurcation branches.

Free Vibration and Tension Buckling of Circular Plates With Diametral Point Forces

1990 ◽  
Vol 57 (4) ◽  
pp. 995-999 ◽  
Author(s):  
E. F. Ayoub ◽  
A. W. Leissa
Keyword(s):  
Circular Plate ◽  
Tensile Force ◽  
Ritz Method ◽  
Compressive Force ◽  
Free Vibrations ◽  
Plane Elasticity ◽  
Mode Shapes ◽  
Simply Supported ◽  
Symmetry Classes ◽  
Tensile Forces

This paper presents the first known results for the free vibrations of a circular plate subjected to a pair of static, concentrated forces acting on the boundary at opposite ends of a diameter. The closed-form exact solution of the plane elasticity problem is used to provide the in-plane stress distribution for the vibration problem. A proper procedure using the Ritz method is developed for solving the latter problem for clamped, simply supported, or free boundary conditions. Numerical results are given for the vibration frequencies of a simply supported circular plate, which separate into four symmetry classes of mode shapes. Compressive buckling loads for each symmetry class are determined as a special case as the frequencies decrease to zero with increasing compressive force. Tracking the frequency versus loading data with increasing tensile forces shows that buckling due to tensile force can also occur, and the critical value of the force is found.

The Behaviour of a Clamped Circular Plate in Compression

1961 ◽  
Vol 12 (1) ◽  
pp. 51-64 ◽  
Author(s):  
A. N. Sherbourne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Circular Plate ◽  
Digital Computer ◽  
Large Deflection ◽  
Theoretical Solution ◽  
Uniform Compression ◽  
Simply Supported ◽  
Von Karman

SummaryA theoretical solution is presented for the problem of the clamped circular plate loaded in uniform compression. The solution employs a numerical method programmed for a digital computer. Instead of solving the classical von Kármán large deflection equations, a step-by-step integration of the elastic differential equations of equilibrium is carried out until suitable boundary conditions are attained. The method is an extension of one developed earlier to explain the behaviour of the simply-supported plate.

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