Analytical investigation of the dynamic response of a Timoshenko thin-walled beam with asymmetric cross section under deterministic loads

Ali Akbari Rasa, Ahmad (2017) Analytical investigation of the dynamic response of a Timoshenko thin-walled beam with asymmetric cross section under deterministic loads. Masters thesis, university of Mohaghegh Ardabili.

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Abstract

One of the main structural members which has a broad use in structure design, including aerospace, mechanic and construction structures is thin-walled beam. Some researches about theoretical analysis of uniform beam dynamic response with coupled bending-torsional vibration affected by fixed and random loads have been carried out. In most of these researches, cross-section of the beam was mono symmetric. Moreover, to model beam bending, Euler-Bernoulli theory is not capable of producing correct results when beams with large sections compared to their lengths or extraction of natural frequencies of higher modes are under study. In such conditions, Timoshinko beam theory in which shear deformation parameters and rotary inertia are considered should be employed. Thus, it is tried to consider modes where thin-walled Timoshenko beam is completely unsymmetrical and the response is delivered theoretically by using modal analysis. In the first step, differential equations of motion of the mentioned beam are extracted by using Hamilton's principle. The obtained equations are five dependent partial differential equations. The dependence of vibratory flexural and torsional components of the beam is because of non-concurrency of centroid and shear centers. Timoshenko theory has been used for modeling beam bending and Vlasov torsion theory has been used for modeling torsional twisting. Dynamic features of the beam are obtained by using free vibration analysis. For this purpose, the dynamic stiffness matrix is used. Applying the dynamic stiffness matrix method on the movement differential equations leads to the issue of nonlinear special value where Wittrick-Williams algorithm is employed to solve it. Orthogonality property of vibratory modes are extracted by applying differential equations of motion. Then the dynamic response of the beam with coupled bending-torsional vibrations affected by fixed loads with constant harmonic and step distribution is obtained. Finally, numerical examples are provided in order to determine the validation and verification of the mentioned formularizations.

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