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.

References

1.
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
.10.1177/1045389X05047598
2.
Ayadoğlu
,
M.
,
2008
, “
Semi-Inverse Method for Vibration and Buckling of Axially Functionally Graded Beams
,”
J. Reinf. Plast. Compos.
,
27
(
7
), pp.
683
691
.10.1177/0731684407081369
3.
Ece
,
M. C.
,
Ayadoğlu
,
M.
, and
Taskin
,
V.
,
2007
, “
Vibration of a Variable Cross-Section Beam
,”
Mech. Res. Commun.
,
34
(
1
), pp.
78
84
.10.1016/j.mechrescom.2006.06.005
4.
Calio’
,
I.
, and
Elishakoff
,
I.
,
2005
, “
Closed-Form Solutions for Axially Graded Beams-Columns
,”
J. Sound Vib.
,
280
(
3
), pp.
1083
1094
.10.1016/j.jsv.2004.02.018
5.
Li
,
Q. S.
.,
2009
, “
Exact Solution for the Generalized Euler's Problem
,”
ASME J. Appl. Mech.
,
76
(
4
), p.
041015
.10.1115/1.2937151
6.
Maroti
,
G.
,
2011
, “
Finding Closed Form Solutions of Beam Vibrations
,”
Pollack Period.
,
6
(
1
), pp.
141
154
.10.1556/Pollack.6.2011.1.13
7.
Calio’
,
I.
,
Gladwell
,
G. M. L.
, and
Morassi
,
A.
,
2011
, “
Families of Beams With a Given Buckling Spectrum
,”
Inverse Probl.
,
27
, p.
045006
.10.1088/0266-5611/27/4/045006
8.
Bokaian
,
A.
,
1988
, “
Natural Frequencies of Beam Under Compressive Axial Loads
,”
J. Sound Vib.
,
126
(
1
), pp.
49
65
.10.1016/0022-460X(88)90397-5
9.
Elishakoff
,
I.
,
2005
,
Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions
,
CRC Press
,
Boca Raton, FL
.
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