The behavior of a beam fixed on small sets of one of its extremities

Abstract

In this paper we study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity system in a thin cylinder (a beam). The beam is fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities but only on several small fixing sets on the other extremity; on the remainder of the boundary the Neumann boundary condition holds. As far as the boundary conditions are concerned, the result depends on the size and on the arrangement of the small fixing sets. In particular, we show that it is equivalent to fix the beam at one of its extremities on 3 unaligned small fixing sets or on 1 or 2 fixing set(s) of bigger size.