Natural frequencies of sandwich beams including shear and rotary effects

Abstract

Abstract The Bernoulli-Euler model of the vibrating beam can be used in the analysis of a three-layered sandwich beam when rotary and shear effects are neglected. When these effects are included a different model must be used, depending upon the thickness of the faces. Thin faces act as membranes and carry no shear. Thick faces act as beams and carry a part of the shear along with the core. In both cases the bending stiffness must be supplied by the faces, which are composed of material that is highly resistant to deformation. A thin disk beam-shaped body in plane stress has been proposed as a model. The independent variables describing the deformations are the longitudinal and transverse displacements. These displacement functions are unknown functions of the spatial co-ordinates and time. They are determined by a variational analysis using Hamilton’s principle, which furnishes two partial differential equations along with the natural boundary conditions. The displacement functions are constructed so as to satisfy the condition of zero tractions on the faces and at the same time permit warping due to bending. A closed form solution for a simply supported beam is available. Because of the extreme difficulty experienced in finding solutions of the Euler equations for other cases, a Galerkin type solution is used to obtain approximate eigenvalues. The studies presented show that increasing the thickness of the faces does not necessarily lead to an increase in the natural frequencies of end supported beams. The homogeneous case is also studied and its relation to the sandwich beam is set forth by comparing the corresponding eigenvalues.