Application of the boundary-domain element method to the pre/post-buckling problem of von Kármán plates

1999 ◽  
Vol 23 (5-6) ◽  
pp. 399-404 ◽  
Author(s):  
Masataka Tanaka ◽  
Toshiro Matsumoto ◽  
Zhudong Zheng
Keyword(s):  
Von Karman ◽  
Post Buckling ◽  
Von Kármán ◽  
Element Method

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.

scholarly journals Comparison of the Von Kármán and Kirchhoff models for the post-buckling and vibrations of elastic beams

2021 ◽  
Author(s):  
Sébastien Neukirch ◽  
Morteza Yavari ◽  
Noël Challamel ◽  
Olivier Thomas
Keyword(s):  
Elastic Beams ◽  
Buckled Beam ◽  
Von Karman ◽  
Order Expansion ◽  
Validity Range ◽  
Post Buckling ◽  
International Audience ◽  
Von Kármán ◽  
Kirchhoff Model ◽  
General Scope

International audience We compare different models describing the buckling, post-buckling and vibrations of elastic beams in the plane. Focus is put on the first buckled equilibrium solution and the first two vibration modes around it. In the incipient post-buckling regime, the classic Woinowsky-Krieger model is known to grasp the behavior of the system. It is based on the von Kármán approximation, a 2nd order expansion in the strains of the buckled beam. But as the curvature of the beam becomes larger, the Woinowsky-Krieger model starts to show limitations and we introduce a 3rd order model, derived from the geometrically-exact Kirchhoff model. We discuss and quantify the shortcomings of the Woinowsky-Krieger model and the contributions of the 3rd order terms in the new model, and we compare them both to the Kirchhoff model. Different ways to nondi-mensionalize the models are compared and we believe that, although this study is performed for specific boundary conditions, the present results have a general scope and can be used as abacuses to estimate the validity range of the simplified models.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

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