Static and Dynamic Analysis of Shells

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Homogeneous, isotropic and linear elastic Vlasov-Reissner thin shallow shells under static and dynamic conditions are numerically analysed with the aid of the boundary element method and its variations. Static analysis of shallow shells can first be accomplished by the conventional direct or indirect boundary element method, which employs the fundamental solution of the problem and is based on the displacement or the fiexural displacement — membrane stress function formulation of shallow shells. This method which requires only a boundary discretization, is not efficient for the general case due to the complexity of the fundamental solution and only for the special cases of spherical and circular cylindrical shells, for which simpler fundamental solutions exist, may be advantageous. A better approach is the direct domain/boundary element method, which employs the plate fundamental solutions in flexure and stretching in its formulation. This creates boundary as well as domain integrals due to the flexure-stretching coupling terms and thus requires a boundary as well as an interior discretization. However, the simplicity of the fundamental solution results in a more efficient and general scheme. Free and forced vibrations of shallow shells are also treated by the direct domain/boundary element method, which employs the static fundamental solutions of a plate in flexure and stretching. This creates domain integrals due to the flexure-stretching coupling terms as well as due to the inertia terms. Transient forced vibrations are analysed with the aid of a time marching scheme in the time domain or Laplace transform with respect to time, in which case a numerical inversion of the transformed solution is required to obtain the time domain response. The effect of external viscous or internal viscoelastic damping on the response is also studied. Various numerical examples are presented for illustrating the aforementioned boundary element approaches and some comparisons against analytical methods and the finite element method are made to point out their advantages.