Numerical computation of effective stiffness for spatially graded beam-like structures based on asymptotic homogenization

This work investigates asymptotic homogenization method (AHM) for axially graded beams that are mapped from periodic ones. The unit cell problems, the homogenized constitutive and governing equations are first established theoretically. Then a novel FE formulation of unit cell problems and effective stiffness, distinct from that of the periodic beams, is derived and resolved for solid elements. Besides, to improve analysis efficiency, an updated concise formulation is acquired for shell elements with proper handle of in-plane rotational DOFs, and a MATLAB code is presented to show implementation details. At last, four numerical examples show the correctness of the proposed method.