Keywords: flexural-torsional, vibration, beam, nonuniform torsion, dynamic analysis, warping, flexural, bar, twist, boundary element method, shear deformation.

In engineering practice, we often come across the analysis of beam structures subjected to vibratory loading. This problem becomes much more complicated in the case when the cross section's centroid does not coincide with its shear center (asymmetric beams), leading to the formulation of the flexural-torsional vibration problem. Also, composite structural elements consisting of a relatively weak matrix material reinforced by stronger inclusions or of materials in contact are of increasing technological importance. Steel beams or columns totally encased in concrete, fiber-reinforced materials or concrete plates stiffened by steel beams are the most common examples. Moreover, unless the beam is very "thin" the error incurred from the ignorance of the effect of shear deformation may be substantial, particularly as regards natural frequencies. In this investigation, an integral equation technique is developed for the solution of the aforementioned problem. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli but the same Poisson's ratio and are firmly bonded together. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The solution method is based on the concept of the analog equation. According to this method, the three coupled fourth order hyperbolic partial differential equations are replaced by three uncoupled ones subjected to fictitious time dependent load distributions under the same boundary conditions. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the analog equation method, a boundary element based method. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows.

1. The proposed method can be applied to beams having an arbitrary composite constant cross section and not to a necessarily thin-walled one.
2. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not necessarily coincide with the principal bending one.
3. For the first time in the literature shear deformation effect is taken into account on the flexural-torsional vibration problem of a beam of a non-symmetric constant cross section.
4. Both rotary and warping inertia are taken into account.
5. The beam is supported by the most general linear boundary conditions including elastic support or restrain.
6. The shear deformation coefficients are evaluated using an energy approach, instead of Timoshenko's and Cowper's definitions, for which several authors have pointed out that one obtains unsatisfactory results or definitions given by other researchers, for which these factors take negative values.
7. The proposed method employs a pure boundary element approach (requiring only boundary discretization) resulting in line or parabolic elements instead of area elements of the finite element solutions (requiring the whole cross section to be discretized into triangular or quadrilateral area elements), while a small number of line elements are required to achieve high accuracy.

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