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An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Banerjee, J. R. and Ananthapuvirajah, A. (2018). An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories. International Journal of Mechanical Sciences, 150, pp. 337-347. doi: 10.1016/j.ijmecsci.2018.10.012

Abstract

An exact dynamic stiffness matrix for a beam is developed by integrating the Rayleigh–Love theory for longitudinal vibration into the Timoshenko theory for bending vibration. In the formulation, the Rayleigh–Love theory accounted for the transverse inertia in longitudinal vibration whereas the Timoshenko beam theory accounted for the effects of shear deformation and rotating inertia in bending vibration. The dynamic stiffness matrix is developed by solving the governing differential equations of motion in free vibration of a Rayleigh–Love bar and a Timoshenko beam and then imposing the boundary conditions for displacements and forces. Next the two dynamic stiffness theories are combined using a unified notation. The ensuing dynamic stiffness matrix is subsequently used for free vibration analysis of uniform and stepped bars as well as frameworks through the application of the Wittrick–Williams algorithm as solution technique. Illustrative examples are given to demonstrate the usefulness of the theory and some of the computed results are compared with published ones. The paper closes with some concluding remarks.

Publication Type: Article
Additional Information: © 2018 Elsevier Ltd. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Publisher Keywords: Rayleigh–Love bar, Timoshenko beam, Dynamic stiffness method, Wittrick–Williams algorithm
Subjects: Q Science > QC Physics
T Technology > TJ Mechanical engineering and machinery
Departments: School of Mathematics, Computer Science & Engineering > Engineering > Mechanical Engineering & Aeronautics
URI: http://openaccess.city.ac.uk/id/eprint/20916
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