In this present study, an analysis for the out-of-plane vibration frequencies for different angles (0°, 45°, 60°, 90°) of fibers along the axis of a circular orthotropic beam will be given. The effect of shear deformation and rotary inertia are taken into account. The Gâteaux differential method (GDM) is employed to construct the functional and variational method [1] is applied to obtain the mixed finite element formulation [2]. The governing equations for out-of-plane scalar vibration can be found from the literature [3], in which, there are six variables including a shear force, a bending moment, a torsional moment, two components of rotation and a component of displacement. Field equations can be written in operator form and are given explicitly in matrix form of the operator. Therefore the functional corresponding to the field equations is obtained [2]. In dynamic analysis, the problem of the natural frequencies of free vibrating structural system reduces to the solution of the standard eigenvalue problem. To derive the finite element formulations, first the interpolation functions, by which all unknown internal quantities have been expressed, must be chosen and then are inserted into the functional. After extremization of this functional with respect to twelve nodal variables, the twelve element equations obtained are solved. A computer program is coded in FORTRAN to obtain the natural frequencies. The results are compared with those obtained using ANSYS.

All kinds of boundary conditions can be analyzed using the proposed method. This formulation can be used for wide and relatively thin structures due to the fact that this formulation is free from locking effects. Because of the properties of the new functional obtained by the GDM, it is possible to generate element matrices easily. An agreement is achieved as comparing the results obtained using ANSYS and the FORTRAN computer program.

1

J.D. Oden, J.N. Reddy, "Variational Method in Theoretical Mechanics", Springer, Berlin, 1976.

2

A.Y. Akoz, F. Kadioglu, "The mixed finite element solution of circular beam on elastic foundation", Computers & Structures, 60(4), 643-651, 1996. doi:10.1016/0045-7949(95)00418-1

3

V. Yildirim, "Common effects of the rotary inertia and shear deformation on the out-of-plane natural frequencies of composite circular bars", Composites: Part B, 32, 687-695, 2001. doi:10.1016/S1359-8368(01)00040-3

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