Junction of elastic plates and beams (Preliminary version)

In this paper we study the contact problem for small and large deflections of thin elastic plate, the lateral displacement of which are constrained by a presence of a unilateral support. The problems considered here are systems of nonlinear variational inequalities in transverse displacement w and stress ψ of points on the middle plane of the plate. The unilateral constraint has the form w ≥ −b, where b is the initial distance between the middle plane and rigid frictionles plane below the plate. Such kind of problems belong to the general area of nonsmooth and contact mechnics .

Journal of Mathematical Sciences, 2016

We propose a one-dimensional asymptotic model of an L-shaped junction of two thin two-dimensional elastic beams subject to boundary conditions of different type at external bean ends. The beams are described by standard systems of the Kirchhoff ordinary differential equations, whereas the transmission conditions on coincident (internal) nodes essentially depend on the type of boundary conditions on the external beam ends. The transmission conditions are obtained by analyzing the boundary layer near internal nodes. The asymptotics is justified with the help of a weighted anisotropic Korn inequality. Bibliography: 28 titles. Illustrations: 5 figures.

Journal of Functional Analysis, 1989

We consider a problem in three-dimensional linearized elasticity, posed over a domain consisting of a plate with thickness 2s inserted into a solid whose Lami: constants and density are independent of E. If the Lame constants of the material constituting the plate vary as se3 and its density as a-', we show that the solutions of the three-dimensional eigenvalue problem converge, as E approaches zero, to the solutions of a "coupled, " "pluri-dimensional" eigenvalue problem of a new type, posed simultaneously over a three-dimensional open set with a slit and a twodimensional open set. ri3 1989 Academic Press, Inc.

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.

Mathematical Methods in the Applied Sciences, 2015

Communicated by W. L. Wendland An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces.

Symmetry

We present two new models for dynamic beams deduced from three dimensional theory of linear elasticity. The first model is deduced from virtual work considered for small beam sections. For the second model, we suppose a Taylor-Young expansion of the displacement field up to the fourth order in transverse dimensions of the beam. We consider the Fourier series expansion for considering Neumann lateral boundary conditions together with dynamical equations, we obtain a system of fifteen vector equations with the fifteen coefficients vector unknown of the displacement field. For beams with two fold symmetric cross sections commonly used (for example circular, square, rectangular, elliptical…), a unique decomposition of any three-dimensional loads is proposed and the symmetries of these loads is introduced. For these two theories, we show that the initial problem decouples into four subproblems. For an orthotropic material, these four subproblems are completely independent. For a monoclin...

Journal of Applied Mathematics and Mechanics, 1989

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2013

The analysis of problem of joined elastic beams is presented in comparison with the engineering and asymptotic approaches. Our analysis is based on three-dimensional elasticity theory model and recently developed method of local perturbation , which seems to be an effective tool for analysis of fields in the vicinity of joint. We demonstrate that the method of local perturbation developed in for scalar Laplace equation can be modified for vectorial elasticity theory problem. We demonstrate that the elasticity theory problem in joined domains of small diameter can be decomposed into one-dimensional problem describing global deformation of a system of joined beams and three-dimensional problems describing local deformation of singular joints in uniform fields. The first problem is the classical one, which ignores individual properties of joint absolutely. The second problem initiates reminiscence about the cellular problem of the homogenization theory for periodic structure. In spite of some similarities, the mentioned problems differ significantly. In particular, the joint of normal type (the joint similar in dimensions and material characteristics to the joined beams) does not manifest itself on global level. Due to the strong localization of perturbation of solution, computation of local strains and stresses in the vicinity of joint can be realized with standard FEM software.

International Journal of Mathematics and Mathematical Sciences, 1993

This paper concerns the existence and uniqueness of equilibrium states of a beamcolumn with hinged ends which is acted upon by axial compression and lateral forces and is in contact with a semi-infinite medium acting as a foundation. The problem is formulated as a fourthorder nonlinear boundary value problem in which the source of the nonlinearity comes from the lateral constraint (the foundation). Treating the equation of equilibrium as a nonlinear eigenvalue problem we prove the existence of a pair of eigenvalue/eigenfunction for each arbitrary prescribed energy level. Treating the equilibrium equation as a nonlinear boundary value problem we prove the existence and uniqueness of solution for a certain range of the acting axial compression force. KEY WORDS AND PHRASES. Existence of equilibrium states, beam-column, elastic beam, fourth-order nonlinear boundary value problem, nonlinear eigenvalue problems, variational methods. 1991 AMS SUBJECT CLASSIFICATION CODES. 49G99, 73H05, 73K15.

Journal de Mathématiques Pures et Appliquées, 2007

We consider a set of elastic rods periodically distributed over a 3d elastic plate (both of them with axis x 3 ) and we investigate the limit behavior of this problem as the periodicity ε and the radius r of the rods tend to zero (see below). We use a decomposition of the displacement field in the rods of the form u = U + u where the principal part U is a field which is piecewise constant with respect to the variables (x 1 , x 2 ) (and then naturally extended on a fixed domain), while the perturbation u remains defined on the oscillating domain containing the rods. We derive estimates of U and u in term of the total elastic energy. This allows to obtain a priori estimates on u without solving the delicate question of the dependence, with respect to ε and r, of the constant in Korn's inequality in such an oscillating domain. To deal with the field u, we use a version of an unfolding operator which permits both to rescale all the rods and to work on the same fixed domain as for U to carry out the homogenization process. The above decomposition also helps in passing to the limit and to identify the limit junction conditions between the rods and the 3d plate. même domaine fixe que pour U afin d'analyser le problème d'homogénéisation. La décomposition ci-dessus facilite aussi le passageà la limite et l'obtention les conditions de jonction limites entre les poutres et la plaque 3d.