Beam Response Derived from 3 D Hybrid BoundaryIntegral Method in Elastodynamics

The aim of the present paper is to associate a new symmetric boundary integral method to well known symmetric domain methods with the purpose of improving solutions of multi-eld problems with coupled boundary and domain methods. From a single-eld variational principle for a 3D linear, elastodynamic state a three-eld hybrid principle is developed by Hamiltons principle and decoupling displacements in the domain from those on the boundary. Compatibility is enforced in a weak sense. For investigating steady-state vibrations the functional is transformed in the frequency domain. Superimposed singular fundamental solutions of the Lam e-Navier eld equations generated by Dirac functions and weighted by ctitious loads are used as test functions in the domain. In the absence of body forces they cancel the remaining domain integral in the hybrid principle and lead to a boundary integral formulation. The boundary variables are discretized by boundary elements. A symmetric dynamic stiiness matrix equation is gained which relates nodal displacements and tractions on the boundary. An application of the hybrid boundary integral method is derived. Because acoustic and hydroacous-tic radiation in 2D is predominantly generated by bending waves, the 3D hybrid method is adopted for 1D beams. As the boundary of a nite beam degenerates to two nodes no shape functions are needed. This is why the theory is shown to give analytical results of dynamical beam analysis.