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Shock and Vibration
Volume 2018, Article ID 1348970, 12 pages
Research Article

A Finite Element Formulation for Bending-Torsion Coupled Vibration Analysis of Delaminated Beams under Combined Axial Load and End Moment

Department of Aerospace Eng., Ryerson University, Toronto, ON, M5B-2K3, Canada

Correspondence should be addressed to Seyed M. Hashemi; ac.nosreyr@mehsahms

Received 29 May 2018; Accepted 19 August 2018; Published 25 September 2018

Academic Editor: Giosuè Boscato

Copyright © 2018 Mir Tahmaseb Kashani and Seyed M. Hashemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Free vibration analysis of beams with single delamination undergoing bending-torsion coupling is made, using traditional finite element technique. The Galerkin weighted residual method is applied to convert the coupled differential equations of motion into to a discrete problem, where, in addition to the conventional mass and stiffness matrices, a delamination stiffness matrix, representing the extra stiffening effects at the delamination tips, is introduced. The linear eigenvalue problem resulting from the discretization along the length of the beam is solved to determine the frequencies and modes of free vibration. Both “free mode” and “constrained mode” delamination models are considered in formulation, and it is shown that the continuity (both kinematic and force) conditions at the beam span-wise locations corresponding to the extremities of the delaminated region, in particular, play a great role in “free mode” model formulation. Current trends in the literature are examined, and insight into different types of modeling techniques and constraint types are introduced. In addition, the data previously available in the literature and those obtained from a finite element-based commercial software are utilized to validate the presented modeling scheme and to verify the correctness of natural frequencies of the systems analyzed here. The paper ends with general discussions and conclusions on the presented theories and modeling approaches.