Variational principles for stiffnesses of a non-homogeneous beam

General variational principles and two-sided estimates for a nonhomogeneous beam of periodic structure are derived on the basis of the asymptotic method and analysis of Lagrange and Castigliano functionals for the so-called cell problem. The nonhomogeneity of the beam can be a result of nonhomogeneity of the material the beam is made of (which is also possible for monolithic bodies), and nonhomogeneity of the beam geometry (this case is impossible for monolithic bodies). The present consideration covers both these cases.