Computation of Unilaterally Supported Elastic Plates

Nonlinear Analysis: Real World Applications, 2007

The unilateral contact problem for the von Kármán plate including postbuckling is numerically studied in this paper. The mathematical model consists of a system of nonlinear inequalities and equations for the transversal displacements and the stress function on the middle plane of the plate, respectively. The boundary conditions correspond to simply supported or partially clamped plates. The lateral displacements are constrained by the presence of a rigid support. A variational principle with penalty is used to treat the mechanical model. Then the variational penalized problem is solved by a spectral method. For the obtained discrete model we develop an iterative scheme based on Newton's iterations, combined with numerical continuation coupled with an appropriate procedure for the choice of the penalty and regularization parameters. Numerical results demonstrate the effectiveness of the proposed method. ᭧

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1995

Contact of a Plate and an Elastic Body In der vorliegenden Arbeit wird das Problem des Kontaktes zwischen einer Platte und einem elastischen Korper betrachtet, wobei die Platte am elastischen Korper bleibt. Das Problem wurde in [ I ] untersucht. wobei die Autoren die Boussinesq-Losung verwendeten, um die Verriickungen der Randpunkte der elastischen Grundlage darzustellen, die in ihrem Fall ein homogener elastischer Halbraum war. Wir verwenden hier fur unsere Modellierung die z. B. in [a] gegebene lineare Elastizitatstheorie. Der Kontakt zwischen der Platte und dem elastischen Korper wird hier durch eine Signorini-Bedingung gesteuert und somit kann das Problem auf eine eventuell nicht-koerzitive Variationsungleichung reduziert werden. Wir befassen uns hier mit der Existenz und Eindeutigkeit einer Losung dieser Ungleichung. AuPerdem geben wir eine Finite-Elemente-Approximation dieser Losung. In this paper we consider the problem of contact between a plate and an elastic body, the plate resting on the elastic body. The problem was studied in [ I ] where the authors used the Boussinesq solution to represent the displacement of boundary points of the elastic foundation which in their case is a homogeneous elastic half space. We use for our modeling here the linear elasticity theory given e.g. in [a]. The contact between the plate and the elastic body is here governed by a Signorini condition and the problem can thus be reduced to a variational inequality that may be noncoercitive. We are concerned here with the existence and uniqueness of a solution ofthis inequality. We also present ajinite element approximation of its solutions.

A numerical methodology for the analysis of plates, governed by Kirchhoff or Mindlin theory, on a tensionless (and Motionless) Winkler foundation or elastic half-space is presented. The finite element method is used for the plate and foundation, and the contact problem is dealt with directly as a minimization problem, involving only the original variables, subjected to inequality constraints. The problem is transformed into a standard linear complementarity problem that is solved by the Lemke's algorithm. Once the LCP is solved, the contact surface and foundation reaction can be obtained The method is illustrated by three numerical examples and the results are compared with available analytical and numerical results.

1998

This work aims at developing a numerical methodology for analysis of plates with contact constraints (unilateral and bonded). The unilateral contact is due to the presence of tensionless supports and foundations. The finite element method is used for the plate and foundation, and the contact problem is dealt with directly as a minimisation problem, involving only the original variables, subjected to inequality constraints. Then, the resulting linear programming problem is solved by the Lemke's algorithm. The results of a numerical investigation relative to three rectangular plates are presented in this paper.

2008

We consider the problem of unilateral contact between two elastic perpendicular plates. The main focus is on the boundary conditions along the contact zone. We propose a mixed domain formulation. Some limit cases for the considered problem are justified. In particular, a unilateral contact between a plate and a beam is also analyzed.

Siberian Mathematical Journal, 2006

We consider the variational free boundary problem describing the contact of an elastic plate with a thin elastic obstacle. The contact domain is unknown a priori and should be determined. The problem is described by a variational inequality for a fourth-order operator. The constraint on the displacement is given on a set of dimension less than that of the solution domain. We find the boundary conditions on the set of the possible contact and their exact statement. We justify the mixed statement of the problem and analyze the limit cases corresponding to the unbounded increase of the elasticity coefficients of the contacting bodies.

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, 1984

2014

In this paper we apply an energy method to examine the axisymmetric contact problem for a flexible circular plate in smooth contact an incompressible elastic halfspace, where the linear elastic shear modulus varies exponentially with depth. The approach adopted approximates the deflected shape of the plate by a power series expansion which satisfies the kinematics of deformation of the plate and the Kirchhoff boundary condition at the edge of the plate. The coefficients in the series are evaluated by making use of the principle of minimum potential energy. Results are obtained for the maximum deflection, the relative deflection and the maximum flexural moment in the circular plate. The results derived from the proposed procedure are compared with equivalent results derived from a computational procedure.

International Journal for Numerical Methods in Engineering, 1996

The present work investigates the unilateral frictionless contact between a plate and a rigid obstacle. Two sets of problems are studied: a plate constrained through unilateral edge supports and a plate seating in its undeformed configuration at a given distance from a rigid support. The attention is concentrated on two augmented Lagrangian formulations. The algorithmic implementation within a finite-element scheme is presented and discussed. The importance of using appropriate plate elements for the discretization of the structure is stressed. New gap elements compatible with a robust plate element are derived. Computational aspects are emphasized. A simple and effective numerical integration for the determination of the gap stiffnesses in partial contact with the support is proposed. Numerical results are carried out and compared with analytical solutions. The convergence to the solution of the perfectly constrained problem is numerically investigated. The inadequacy of the penalty method and the satisfactory performance obtained from only one augmented Lagrangian procedure are emphasized.

1980

The axially symmetric flexural interaction of a uniformly loaded circular elastic plate resting in smooth contact with a transversely isotropic elastic halfspace is examined by using a variational method.