No abstract

An exact analytical solution of a harmonically vibrating beam-column element resting on an elastic Winkler foundation is derived. The solution covers four cases comprised of constant compressive or tensile axial force with the restrictions ks – mω2 < 0 and ks – mω2 > 0. The proposed solution is not restricted to a particular range of magnitudes of the foundation parameter. Closed form solutions of dynamic shape functions are explicitly derived for each sub-case to obtain frequency-dependent dynamic stiffness terms that constitute the exact dynamic stiffness matrices. Four numerical examples are provided to demonstrate the merits of the present study.

In this paper, we have elaborated on the decomposition methods such as irreducible decomposition, orthonormal tensor basis, harmonic and spectral decomposition for elastic constant tensor. Irreducible decomposition and orthonormal tensor basis methods are developed by using the results of existing theories in the literature. As examples to each decomposition method, we give results for the decomposition of elastic constant tensor in triclinic symmetry as well as materials with isotropic and transversely isotropic symmetry. Numerical examples serve to illustrate and verify each of the four decomposition methods. These examples are used to compare the decomposition methods explicitly. As a result of comparison process, it is stated that the spectral method is a non-linear invariant decomposition method that yields non-linear orthogonal parts contrary to the other three methods which are linear invariant decomposition methods. It is also shown that total scalar (isotropic) part is decomposed into two physically meaningful orthogonal parts by irreducible decomposition, orthonormal tensor basis and spectral methods. While in harmonic decomposition method, decomposition of total scalar part is not orthogonal. We propose that it is possible to make these parts orthogonal to each other.