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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 108
Edited by: J. Kruis, Y. Tsompanakis and B.H.V. Topping
Paper 105

Derivation of the Dynamic Stiffness Matrix of a Functionally Graded Beam using Higher Order Shear Deformation Theory

H. Su1 and J.R. Banerjee2

1University of Northampton, United Kingdom
2City University London, United Kingdom

Full Bibliographic Reference for this paper
H. Su, J.R. Banerjee, "Derivation of the Dynamic Stiffness Matrix of a Functionally Graded Beam using Higher Order Shear Deformation Theory", in J. Kruis, Y. Tsompanakis, B.H.V. Topping, (Editors), "Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 105, 2015. doi:10.4203/ccp.108.105
Keywords: free vibration, functionally graded beams, dynamic stiffness method, Wittrick-Williams algorithm, parabolic shear deformation beam theory.

The dynamic stiffness matrix of a functionally graded beam (FGB) is developed using a higher order shear deformation theory. The material properties of the FGB are varied in the thickness direction based on a power-law. The kinetic and potential energies of the beam are formulated by accounting for a parabolic shear stress distribution. Hamilton's principle is used to derive the governing differential equations of motion in free vibration. The analytical expressions for axial force, shear force, bending moment and higher order moment at any cross-section of the beam are obtained as a by-product of the Hamiltonian formulation. The differential equations are solved in closed analytical form for harmonic oscillation. The dynamic stiffness matrix of the FGB is then constructed by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is applied to the dynamic stiffness matrix of the FGB to compute its natural frequencies and mode shapes in the usual way after solving the eigenvalue problem. Finally, some conclusions are drawn.

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