Microstructure Theory for a Composite Beam

Abstract

A continuum model with microstructure is constructed for laminated beams. In deriving equations, each constituent layer is considered as a Timoshenko beam. With a certain kinematical assumption regarding the deformations of the composite beam the kinetic and strain energies, as well as the variation of the work done by external forces, are computed. The energy and work expressions are “smoothed out” by a smoothing operation, thus transforming the laminated structuring into microstructure of a macro-homogeneous beam. Subsequent application of Hamilton’s principle yields the equations of motion and the boundary conditions. The equations thus obtained are employed to investigate free flexural waves in a composite beam. It is found that the dispersion curves according to the present theory agree very well with the curves obtained according to an exact analysis. Results according to the effective modulus theory are presented for comparison. Two simplified versions of the microstructure beam theory are also developed and discussed.