A bending moment beam eigenelement with piecewise shape functions

The classic eigenelement method is proposed by XING Yufeng and TIAN Jinmei recently. This method can be used to deal with the macroscopic properties such as frequencies and elastic parameters of composites. Compared to conventional homogenization method, the classic eigenelement method calculates the global stiffness of unit cell directly, instead the homogeneous modules. The eigenelement method is more precise than the conventional homogenization method with the same number of elements, and the amount of computational work reduces greatly compared with the ordinary finite element method with a good agreement between them. In order to calculate microscopic properties more precisely, a bending moment beam eigenelement is proposed based on the variational principle, in which the eigen-shape functions are defined in segments, so the discontinuous of materials and geometries is allowed within the beam eigenelement.The bending moment beam eigenelement is derived in details according to the static analysis of the beam cell subjected to a unit bending moment at one end. In order to evaluate the new proposed bending moment beam eigenelement (BBEE), the displacements, stresses and inherent frequencies are computed by using the new beam eigenelement method, ordinary finite element method, the classic eigenelement method and conventional homogenization method respectively. The comparison among the results by different methods, show that the bending moment eigenelement method is more precise than the classic eigenelement method and conventional homogenization method with the same number of elements. The larger the difference of parameters for each subelement in eigenelement is, the larger the advantages of the eigenelement. And the amount of computational work reduces greatly compared to the ordinary finite element method with the same precision.