scholarly journals THIN ELASTIC PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE THE VON KARMAN LIMITS

2021 ◽  
Author(s):  
Sergey Selyugin
Keyword(s):  
Virtual Work ◽  
Plate Thickness ◽  
Geometrically Nonlinear ◽  
Large Deflections ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Virtual Work Principle ◽  
Von Karman ◽  
Von Kármán ◽  
Thin Elastic Plates

Thin elastic plates (homogeneous or composite) experiencing large deflections are considered. The deflections are much larger than the plate thickness. The geometrically nonlinear elasticity theory and the Kirchhoff assumptions are employed. Small elongations and shears are assumed. Following Novozhilov, the strain expressions are derived. Then, under a small in-plane rotation assumption and using the virtual work principle, the equilibrium equations and the boundary conditions are obtained. The equations/conditions become the known von Karman ones for the case of moderate deflections. The solutions of the obtained equations may be used as benchmarks for the nonlinear structural analysis (e.g., FEM) software in the case of large deflections.

scholarly journals A KINEMATIC VARIATIONAL PRINCIPLE FOR THIN METALLIC AND COMPOSITE PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE THE VON KARMAN LIMITS

2021 ◽  
Author(s):  
Sergey Selyugin
Keyword(s):  
Variational Principle ◽  
Plate Thickness ◽  
Composite Plates ◽  
Geometrically Nonlinear ◽  
Large Deflections ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Von Karman ◽  
Von Kármán ◽  
Thin Elastic Plates

Thin elastic plates (metallic or composite) experiencing large deflections are considered. The plate deflections are much larger than the plate thickness. The geometrically nonlinear elasticity theory and the Kirchhoff assumptions are employed. The elongations, the shears and the in-plane rotations are assumed to be small. A kinematic variational principle leading to a boundary value problem for the plate is derived. It is shown that the principle gives proper equilibrium equations and boundary conditions. For moderate plate deflections the principle is transformed to the case of the von Karman plate.

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.

scholarly journals A virtual element method for the von Kármán equations

2021 ◽  
Vol 55 (2) ◽  
pp. 533-560
Author(s):  
Carlo Lovadina ◽  
David Mora ◽  
Iván Velásquez
Keyword(s):  
Optimal Error Estimates ◽  
Transverse Displacement ◽  
Discrete Problem ◽  
Elastic Plates ◽  
Virtual Element Method ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Virtual Element ◽  
Von Kármán ◽  
Element Method

In this article we propose and analyze a Virtual Element Method (VEM) to approximate the isolated solutions of the von Kármán equations, which describe the deformation of very thin elastic plates. We consider a variational formulation in terms of two variables: the transverse displacement of the plate and the Airy stress function. The VEM scheme is conforming inH2for both variables and has the advantages of supporting general polygonal meshes and is simple in terms of coding aspects. We prove that the discrete problem is well posed forhsmall enough and optimal error estimates are obtained. Finally, numerical experiments are reported illustrating the behavior of the virtual scheme on different families of meshes.

Effects of symmetric imperfections on the behavior of bistable struts

2016 ◽  
Vol 22 (12) ◽  
pp. 2240-2252 ◽  
Author(s):  
Jianguo Cai ◽  
Xiaowei Deng ◽  
Jian Feng
Keyword(s):  
Virtual Work ◽  
Variable Geometry ◽  
Equilibrium Equations ◽  
Buckling Mode ◽  
Concentrated Load ◽  
Virtual Work Principle ◽  
Shallow Arches ◽  
Buckling Behavior ◽  
Nonlinear Equilibrium ◽  
Post Buckling

The behavior of a bistable strut for variable geometry structures was investigated in this paper. A three-hinged arch subjected to a central concentrated load was used to study the effect of symmetric imperfections on the behavior of the bistable strut. Based on a nonlinear strain–displacement relationship, the virtual work principle was adopted to establish both the pre-buckling and buckling nonlinear equilibrium equations for the symmetric snap-through buckling mode. Then the critical load for symmetric snap-through buckling was obtained. The results show that the axial force is in compression before the arch is buckled, but it becomes in tension after buckling. Thus, the previous formulas cannot be used for the analysis of post-buckling behavior of three-hinged shallow arches. Then, the principle of virtual work was also used to establish the post-buckling equilibrium equations of the arch in the horizontal and vertical directions as well as the static boundary conditions, which are very important for bistable struts.

A Simplified Approach to the Analysis of Large Deflections of Plates

1974 ◽  
Vol 41 (2) ◽  
pp. 523-524 ◽  
Author(s):  
J. Mazumdar ◽  
R. Jones
Keyword(s):  
Arbitrary Shape ◽  
Horizontal Direction ◽  
Large Deflections ◽  
Elastic Plates ◽  
Elliptical Plate ◽  
Simplified Approach ◽  
Contour Lines ◽  
Thin Elastic Plates

This paper uses the method of constant deflection contour lines [1–3] to analyze the nonlinear large deflections of thin elastic plates of arbitrary shape in a new fashion. As an illustration the case of an elliptical plate with edges constrained against motion in the horizontal direction is discussed.

LARGE DISPLACEMENT BEAM MODEL FOR CREEP BUCKLING ANALYSIS OF FRAMED STRUCTURES

2009 ◽  
Vol 09 (01) ◽  
pp. 61-83 ◽  
Author(s):  
GORAN TURKALJ ◽  
DOMAGOJ LANC ◽  
JOSIP BRNIC
Keyword(s):  
Virtual Work ◽  
Buckling Analysis ◽  
Large Displacement ◽  
Isotropic Hardening ◽  
Beam Model ◽  
Inelastic Behavior ◽  
Equilibrium Equations ◽  
Virtual Work Principle ◽  
Framed Structures ◽  
Spatial Beam

In this work, a one-dimensional beam model for buckling analysis of framed structures under large displacement creep regimes is presented. The equilibrium equations of a prismatic and straight spatial beam element are formulated in the framework of corotational description, using the virtual work principle. Although the translations and rotations of the element are allowed to be large, the strains are assumed to be small. The material of a framed structure is assumed to be homogenous and isotropic. The bilinear elastic–plastic model with isotropic hardening and the power creep law are adopted for describing the inelastic behavior of the material. The numerical algorithm is implemented in a computer program called BMCA and its reliability is validated through test examples.

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