A transversely isotropic magneto-electro-elastic Timoshenko beam model incorporating microstructure and foundation effects

A new model is developed for transversely isotropic magneto-electro-elastic Timoshenko beams by using a variational formulation based on Hamilton's principle. An extended modified couple stress theory and a two-parameter Winkler-Pasternak elastic foundation model are applied to describe the microstructure and foundation effects, respectively. The equations of motion and complete boundary conditions for the beam are simultaneously obtained. The new model includes the models for transversely isotropic piezoelectric and piezo-magnetic Timoshenko beams and the model for transversely isotropic magneto-electro-elastic Bernoulli-Euler beams, all incorporating the microstructure and foundation effects, as special cases. To illustrate the newly developed beam model, the static bending and free vibration problems of a simply supported transversely isotropic magneto-electro-elastic beam subjected to a uniformly distributed load are analytically solved by directly applying the general formulas derived. Numerical results are also provided to qualitatively show the micro-structure, foundation and magneto-electro-elastic coupling effects on the responses of the simply supported beam.