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Research Papers

Some Intriguing Results Pertaining to Functionally Graded Columns

[+] Author and Article Information
Isaac Elishakoff

Fellow ASME
e-mail: elishako@fau.edu

Yohann Miglis

Department of Ocean and Mechanical Engineering,
Florida Atlantic University,
Boca Raton, FL 33431-0991

1Corresponding author.

Manuscript received October 24, 2011; final manuscript received October 19, 2012; accepted manuscript posted October 30, 2012; published online May 23, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 80(4), 041021 (May 23, 2013) (9 pages) Paper No: JAM-11-1395; doi: 10.1115/1.4007983 History: Received October 24, 2011; Revised October 19, 2012; Accepted October 30, 2012

Some intriguing results are reported in conjunction with closed form solutions obtained for a clamped-free vibrating inhomogeneous column under an axial concentrated load using the semi-inverse method. Fourth order polynomial is postulated for both the vibration mode shape and buckling displacement. Solution is provided for the flexural rigidity and the natural frequency. It is shown that, for each level of axial loading, there may exist up to five flexural rigidities satisfying the governing differential equation and boundary conditions.

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References

Elishakoff, I., and Endres, J., 2005, “Extension of Euler's Problem to Axially Graded Columns: Two Hundred and Sixty Years Later,” J. Intell. Mater. Syst. Struct., 16(1), pp. 77–83. [CrossRef]
Ayadoğlu, M., 2008, “Semi-Inverse Method for Vibration and Buckling of Axially Functionally Graded Beams,” J. Reinf. Plast. Compos., 27(7), pp. 683–691. [CrossRef]
Ece, M. C., Ayadoğlu, M., and Taskin, V., 2007, “Vibration of a Variable Cross-Section Beam,” Mech. Res. Commun., 34(1), pp. 78–84. [CrossRef]
Calio’, I., and Elishakoff, I., 2005, “Closed-Form Solutions for Axially Graded Beams-Columns,” J. Sound Vib., 280(3), pp. 1083–1094. [CrossRef]
Li, Q. S.., 2009, “Exact Solution for the Generalized Euler's Problem,” ASME J. Appl. Mech., 76(4), p. 041015. [CrossRef]
Maroti, G., 2011, “Finding Closed Form Solutions of Beam Vibrations,” Pollack Period., 6(1), pp. 141–154. [CrossRef]
Calio’, I., Gladwell, G. M. L., and Morassi, A., 2011, “Families of Beams With a Given Buckling Spectrum,” Inverse Probl., 27, p. 045006. [CrossRef]
Bokaian, A., 1988, “Natural Frequencies of Beam Under Compressive Axial Loads,” J. Sound Vib., 126(1), pp. 49–65. [CrossRef]
Elishakoff, I., 2005, Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, CRC Press, Boca Raton, FL.

Figures

Grahic Jump Location
Fig. 1

Variation of the roots of Eq. (38) with axial load P; (a) roots k1, k3, k4, k5 and (b) rook k2

Grahic Jump Location
Fig. 2

Flexural rigidities and associated mode shapes corresponding to the roots (a) k4 = −0.041411, (b) k1 = −0.01573492764, (c) k2 = −0.006766827430, (d) k3 = 0.001291278483, (e) k5 = 0.01033391077

Grahic Jump Location
Fig. 3

Dependence of the squared natural frequency versus load (a) k = k4, (b) k = k1, (c) k = k2, (d) k = k3, (e) k = k5

Grahic Jump Location
Fig. 4

Vibrating clamped-free beam loaded at its tip

Grahic Jump Location
Fig. 5

Five possible flexural rigidities and displacements for (a) k1, (b) k2, (c) k3, (d) k4, (e) k5

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