A Two-Node Finite Element for Linear Magneto-Electric Laminated Timoshenko Beams

A new finite element is presented for linear magnetoelectric straight laminated beam subject to the assumptions of quasi-steady electromagnetic state. The mechanical model is based upon Timoshenko beam theory to account for shear deformation influences. The electromagnetic stacking sequence is proved to enter the equivalent elastic problem by affecting both the stiffness properties of the beam, in terms of axial and flexural coupling, and by modifying the mechanical boundary conditions as distributed loads. Shape functions are first written for the generalized beam mean-line kinematical quantities in such a way the obtained strain field fulfills the homogeneous governing equations of the equivalent elastic problem. The weak form of the governing equations are then obtained by integrating over the element length the equation of motion of the beam opportunely multiplied by the virtual mean-line axial and transverse displacements and by the virtual cross-sectional rotation. Both the virtual and actual kinematical quantities are then expressed in terms of virtual and actual nodal variables by means of the proposed shape functions. By so doing, the definitions of the element mass and stiffness matrices and of the equivalent force vector are straightforwardly obtained. Lastly, numerical results are presented to assess the soundness of the proposed formulation.