Analytical Investigation of the Dynamic Response of a Timoshenko Thin-Walled Beam with Asymmetric Cross Section Under Deterministic Loads

Elham Ghandi*, Ahmed Ali Akbari Rasa
Faculty of Technical and Engineering, University of Mohaghegh Ardabili, Ardabil, Iran

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© 2017 Ghandi and Rasa.

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address of the corresponding author the Faculty of Technical and Engineering, University of Mohaghegh Ardabili, Ardabil, Iran, Tel: +989141509273, Fax: 984533512904; E-mails:;



The objective of the present paper is to analyze dynamic response of the Timoshenko thin-walled beam with coupled bending and torsional vibrations under deterministic loads. The governing differential equations were obtained by using Hamilton’s principle. The Timoshenko beam theory was employed and the effects of shear deformations, Rotary inertia and warping stiffness were included in the present formulations. Dynamic features of underlined beam are obtained using free vibration analysis.


For this purpose, the dynamic stiffness matrix method is used. Application of exact dynamic stiffness matrix method on the movement differential equations led to the issue of nonlinear eigenvalue problem that was solved by using Wittrick–Williams algorithm . Differential equations for the displacement response of asymmetric thin-walled Timoshenko beams subjected to deterministic loads are used for extracting orthogonality property of vibrational modes.


Finally the numerical results for dynamic response in a sample of mentioned beams is presented. The presented theory is relatively general and can be used for various kinds of deterministic loading in Timoshenko thin-walled beams.

Keywords: Thin-walled beams, Timoshenko beam theory, Exact dynamic stiffness matrix method, Wittrick–Williams algorithm, Deterministic dynamic loads, Asymmetric cross section.