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

Huijuan Su; Ranjan Banerjee

doi:10.4203/ccp.108.105

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

Huijuan Su
, Ranjan Banerjee

Research output: Contribution to Book/Report › Conference Contribution › peer-review

Abstract

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.

Original language	English
Title of host publication	Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing
Editors	J. Kruis, Y. Tsompanakis , B.H.V. Topping
Publisher	Civil-Comp Proceedings
Pages	1-12
Number of pages	12
ISBN (Print)	978-1-905088-63-8
DOIs	https://doi.org/10.4203/ccp.108.105
Publication status	Published - 14 May 2015
Event	Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing - Prague, Czech Republic Duration: 1 Sept 2015 → 4 Sept 2015

Conference

Conference	Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing
Country/Territory	Czech Republic
City	Prague
Period	1/09/15 → 4/09/15

Keywords

Free vibration
functionally graded beams
dynamic stiffness method
Wittrick-Williams algorithm
parabolic shear deformation beam theory

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10.4203/ccp.108.105

Su_Huijuan_2015_Derivation_of_the_Dynamic_Stiffness_Matrix_of_a_Functionally_Graded_Beam_using_Higher_Order_Shear_Deformation_TheoryAccepted author manuscript, 257 KB

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Su, H., & Banerjee, R. (2015). 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 (Eds.), Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing (pp. 1-12). Article 105 Civil-Comp Proceedings. https://doi.org/10.4203/ccp.108.105

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

abstract = "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{\textquoteright}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.",

keywords = "Free vibration, functionally graded beams, dynamic stiffness method, Wittrick-Williams algorithm, parabolic shear deformation beam theory",

author = "Huijuan Su and Ranjan Banerjee",

year = "2015",

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doi = "10.4203/ccp.108.105",

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booktitle = "Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing",

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note = "Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing ; Conference date: 01-09-2015 Through 04-09-2015",

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Su, H & Banerjee, R 2015, Derivation of the Dynamic Stiffness Matrix of a Functionally Graded Beam using Higher Order Shear Deformation Theory. in J Kruis, Y Tsompanakis & BHV Topping (eds), Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing., 105, Civil-Comp Proceedings, pp. 1-12, Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing, Prague, Czech Republic, 1/09/15. https://doi.org/10.4203/ccp.108.105

Derivation of the Dynamic Stiffness Matrix of a Functionally Graded Beam using Higher Order Shear Deformation Theory. / Su, Huijuan; Banerjee, Ranjan.
Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing. ed. / J. Kruis; Y. Tsompanakis ; B.H.V. Topping . Civil-Comp Proceedings, 2015. p. 1-12 105.

Research output: Contribution to Book/Report › Conference Contribution › peer-review

TY - GEN

T1 - Derivation of the Dynamic Stiffness Matrix of a Functionally Graded Beam using Higher Order Shear Deformation Theory

AU - Su, Huijuan

AU - Banerjee, Ranjan

PY - 2015/5/14

Y1 - 2015/5/14

N2 - 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.

AB - 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.

KW - Free vibration

KW - functionally graded beams

KW - dynamic stiffness method

KW - Wittrick-Williams algorithm

KW - parabolic shear deformation beam theory

U2 - 10.4203/ccp.108.105

DO - 10.4203/ccp.108.105

M3 - Conference Contribution

SN - 978-1-905088-63-8

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EP - 12

BT - Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing

A2 - Kruis, J.

A2 - Tsompanakis , Y.

A2 - Topping , B.H.V.

PB - Civil-Comp Proceedings

T2 - Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing

Y2 - 1 September 2015 through 4 September 2015

ER -

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

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