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Keywords: functionally graded beams, third order shear deformation theory, Ritz method.

Summary

Functionally graded materials (FGMs) have many advantages when used in engineering structural components. The methods of FGM fabrication and general information about FGMs including microstructure analysis of the graded materials was presented by Miyamoto et al. [1]. Unlike fiber-matrix laminated composites, FGMs do not have problems of de-bonding and delaminating that result from large inter-laminar stresses. A power law distribution is used to describe the variation of volume fraction of the material composition. With the power law index being not one or zero, the nonlinear compositional profile of material properties can be observed across the functionally graded (FG) beam thickness. Hence, it is expected that an improved third order shear deformation (TSDT) based on the more rigorous kinematics of displacements developed by Shi [2], which presents a cubic variation of displacement through the thickness, will be more suitable to predict free vibration results than the first order shear deformation (FSDT) and classical beam theory (CBT).

The theoretical formulation in the form of an eigenvalue problem derived from the improved theory is solved using the Ritz method with the polynomial displacements and rotation [3,4,5,6]. The numerical frequency results are compared with the published results based on the FSDT with a good agreement [7]. The new frequency results in relation to simply-clamped, simply-free and free-free boundary conditions are also presented in this paper. The significant effects of many aspects such as boundary conditions, material composition and slenderness ratio that cause the change in natural frequencies of FG beams are taken into consideration.

References

1: Y. Miyamoto, W.A. Kaysser, B.H. Rabin, A. Kawasaki, R.G. Ford, "Functionally graded materials: design, processing and application", Kluwer Academic Publishers, London, 1999.
2: G. Shi, "A new simple third-order shear deformation theory of plates", International Journal of Solids and Structures, 44, 4399-4417, 2007. doi:10.1016/j.ijsolstr.2006.11.031
3: M. Aydogdu, "Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method", International journal of Mechanical Sciences, 47, 1740-1755, 2005. doi:10.1016/j.ijmecsci.2005.06.010
4: C.C. Chen, S. Kitipornchai, C.W. Lim, K.M. Liew, "Free vibration of cantilevered symmetrically laminated thick trapezoidal plates", International Journal of Mechanical Sciences, 41, 685-702, 1999. doi:10.1016/S0020-7403(98)00025-3
5: C.S. Huang, A.W. Leissa, "Vibration analysis of rectangular plates with side cracks via the Ritz method", Journal of Sound Vibration, 323, 974-988, 2009. doi:10.1016/j.jsv.2009.01.018
6: S. Kitipornchai, L.L. Ke, J. Yang, Y. Xiang, "Nonlinear vibration of edge cracked functionally graded Timoshenko beams", Journal of Sound Vibration, 324, 962-982, 2009. doi:10.1016/j.jsv.2009.02.023
7: S.A. Sina, H.M. Navazi, H. Haddadpour, "An analytical method for free vibration analysis of functionally graded beams", Materials and Design, 30, 741-747, 2009. doi:10.1016/j.matdes.2008.05.015

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 93 PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: Paper 285 A Theoretical Investigation on the Free Vibration of Functionally Graded Beams N. Wattanasakulpong¹, B.G. Prusty¹, D.W. Kelly¹ and M. Hoffman² ¹School of Mechanical and Manufacturing Engineering, ²School of Material Science and Engineering, University of New South Wales, Sydney, Australia doi:10.4203/ccp.93.285 purchase the full-text of this paper Full Bibliographic Reference for this paper N. Wattanasakulpong, B.G. Prusty, D.W. Kelly, M. Hoffman, "A Theoretical Investigation on the Free Vibration of Functionally Graded Beams", in , (Editors), "Proceedings of the Tenth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 285, 2010. doi:10.4203/ccp.93.285 Keywords: functionally graded beams, third order shear deformation theory, Ritz method. Summary Functionally graded materials (FGMs) have many advantages when used in engineering structural components. The methods of FGM fabrication and general information about FGMs including microstructure analysis of the graded materials was presented by Miyamoto et al. [1]. Unlike fiber-matrix laminated composites, FGMs do not have problems of de-bonding and delaminating that result from large inter-laminar stresses. A power law distribution is used to describe the variation of volume fraction of the material composition. With the power law index being not one or zero, the nonlinear compositional profile of material properties can be observed across the functionally graded (FG) beam thickness. Hence, it is expected that an improved third order shear deformation (TSDT) based on the more rigorous kinematics of displacements developed by Shi [2], which presents a cubic variation of displacement through the thickness, will be more suitable to predict free vibration results than the first order shear deformation (FSDT) and classical beam theory (CBT). The theoretical formulation in the form of an eigenvalue problem derived from the improved theory is solved using the Ritz method with the polynomial displacements and rotation [3,4,5,6]. The numerical frequency results are compared with the published results based on the FSDT with a good agreement [7]. The new frequency results in relation to simply-clamped, simply-free and free-free boundary conditions are also presented in this paper. The significant effects of many aspects such as boundary conditions, material composition and slenderness ratio that cause the change in natural frequencies of FG beams are taken into consideration. References 1 Y. Miyamoto, W.A. Kaysser, B.H. Rabin, A. Kawasaki, R.G. Ford, "Functionally graded materials: design, processing and application", Kluwer Academic Publishers, London, 1999. 2 G. Shi, "A new simple third-order shear deformation theory of plates", International Journal of Solids and Structures, 44, 4399-4417, 2007. doi:10.1016/j.ijsolstr.2006.11.031 3 M. Aydogdu, "Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method", International journal of Mechanical Sciences, 47, 1740-1755, 2005. doi:10.1016/j.ijmecsci.2005.06.010 4 C.C. Chen, S. Kitipornchai, C.W. Lim, K.M. Liew, "Free vibration of cantilevered symmetrically laminated thick trapezoidal plates", International Journal of Mechanical Sciences, 41, 685-702, 1999. doi:10.1016/S0020-7403(98)00025-3 5 C.S. Huang, A.W. Leissa, "Vibration analysis of rectangular plates with side cracks via the Ritz method", Journal of Sound Vibration, 323, 974-988, 2009. doi:10.1016/j.jsv.2009.01.018 6 S. Kitipornchai, L.L. Ke, J. Yang, Y. Xiang, "Nonlinear vibration of edge cracked functionally graded Timoshenko beams", Journal of Sound Vibration, 324, 962-982, 2009. doi:10.1016/j.jsv.2009.02.023 7 S.A. Sina, H.M. Navazi, H. Haddadpour, "An analytical method for free vibration analysis of functionally graded beams", Materials and Design, 30, 741-747, 2009. doi:10.1016/j.matdes.2008.05.015 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £145 +P&P)
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