Keywords: flexural-torsional vibration, beam, boundary element method, bar.

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 where the cross section's centroid does not coincide with its shear center (asymmetric beams), leading to the formulation of the flexural-torsional vibration problem. The extensive use of the aforementioned structural elements necessitates a reliable and accurate analysis of the flexural-torsional vibration problem. Although there is an extensive research on the flexural-torsional vibration analysis of thin-walled beams based on the assumptions of the thin tube theory, using both FEM [1] and BEM [2], to the authors' knowledge, publications on the solution to the general problem do not exist.

In this investigation a boundary element method is developed for the general flexural-torsional vibrations of Euler-Bernoulli beams of arbitrarily shaped cross section. 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 resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. The general character of the proposed method is verified from the fact that all basic equations are formulated with respect to an arbitrary coordinate system, which is not restricted to the principal one. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-tube theory is also investigated.

Let us consider an initially straight Euler-Bernoulli beam of length (Figure 108.1), of constant arbitrary cross-section of area . The homogeneous isotropic and linearly elastic material of the beam cross-section occupies the region of the plane and is bounded by the boundary curves, which are piecewise smooth, i.e. they may have a finite number of corners. In Figure 108.1a, and are coordinate systems (not necessarily principal) through the cross section's centroid and shear center , respectively. Moreover, , are the coordinates of the centroid with respect to system of axes. The beam is subjected to the combined action of the time dependent arbitrarily distributed transverse loading , acting in the and directions, respectively and to the arbitrarily distributed time dependent twisting moment (Figure 108.1b).

The initial boundary value problem of the beam under consideration is described by three coupled partial differential equations inside the beam, subjected to the most general linear boundary conditions associated with the problem at hand and can include elastic support or restrain.

The solution of the problem is achieved employing an efficient and accurate method based on the concept of the analog equation. Several example problems are studied to illustrate the accuracy, the applicability and the efficiency of the proposed method. Moreover, interesting conclusions are drawn for the flexural-torsional vibrations of beams.

1

Kim M.Y., Yun H.T., Kim N.I. "Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beams column", Computers and Structures 81 (2003) 1425-1448. doi:10.1016/S0045-7949(03)00082-8

2

Tanaka M., Bercin A.N., Suzuki R. "Application of the boundary integral equation method to the coupled bending-torsional vibration of elastic beams 20", Engineering Analysis with Boundary Elements (1997) 73-79. doi:10.1016/S0955-7997(97)00059-3

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