A simple design formula is derived here to evaluate the fundamental frequency parameter of initially stressed (subjected to axial concentrated load at the ends) uniform beams resting on elastic foundation. Even though the basis for derivation of the formula is based on the finite element method, the applicability of the formula is general and can be used effectively, once the buckling load parameter, stress free frequency parameter and the applied concentrated load parameter are known, to obtain the fundamental frequency parameter of the stressed beam. The assumption involved in deriving the formula is that the mode shapes of buckling, stress free vibration and stressed vibration are the same. The effectiveness of the formula is demonstrated through numerical examples.

Issue Section:
Technical Briefs
Keywords:
buckling, elasticity, stress-strain relations, finite element analysis, vibrations, matrix algebra
Topics:
Buckling, Design, Finite element methods, Free vibrations, Mode shapes, Stress, Vibration, Finite element analysis, Compression
1.
Hetenyi, M., 1961, Beams on Elastic Foundation, University of Michigan Press, Ann Arbor.
2.
Timoshenko, S. P., and Gere, J. M., 1961, Theory of Elastic Stability, McGraw-Hill, New York.
3.
Zhaohua
,
F.
, and
Cook
,
R. D.
,
1983
, “
Beam Elements on Two Parameter Elastic Foundation
,”
J. Eng. Mech.
,
109
(
6
), pp.
1390
–
1401
.
4.
Shastry
,
B. P.
, and
Rao
,
G. V.
,
1986
, “
A Note on the Vibrations of a Short Cantilever Beam on Elastic Foundation
,”
J. Sound Vib.
,
111
(
1
), pp.
176
–
178
.
5.
Eisenberger
,
M.
, and
Clastornik
,
M.
,
1987a
, “
Vibration and Buckling of a Beam on a Variable Winkler Elastic Foundation
,”
J. Sound Vib.
,
115
(
2
), pp.
233
–
241
.
6.
Eisenberger
,
M.
, and
Clastornik
,
M.
,
1987b
, “
Beams on Variable Two Parameter Elastic Foundation
,”
J. Eng. Mech.
,
113
(
10
), pp.
1454
–
1466
.
7.
Raju
,
K. K.
, and
Rao
,
G. V.
,
1993
, “
Vibrations of Initially Stressed Beams and Plates Around Transition Values of Elastic Foundation Stiffness
,”
J. Sound Vib.
,
161
(
2
), pp.
378
–
384
.
8.
Eisenberger
,
M.
,
1994
, “
Vibration Frequencies for Beam on Variable One and Two Parameter Elastic Foundation
,”
J. Sound Vib.
,
176
(
5
), pp.
577
–
584
.
9.
Naidu
,
N. R.
, and
Rao
,
G. V.
,
1995
, “
Vibrations of Initially Stressed Uniform Beams on a Two Parameter Elastic Foundation
,”
Comput. Struct.
,
57
(
6
), pp.
941
–
943
.
10.
Amba Rao
,
C. L.
,
1967
, “
Effect of End Conditions on the Lateral Frequencies of Uniform Straight Columns
,”
J. Acoust. Soc. Am.
,
42
(
2
), pp.
900
–
901
.
11.
Galef
,
A. E.
,
1968
, “
Bending Frequencies of Compressed Beams
,”
J. Acoust. Soc. Am.
,
44
(
8
), pp.
643
643
.
12.
Bokain
,
A.
,
1988
, “
Natural Frequencies of Beams Under Compressive Axial Loads
,”
J. Sound Vib.
,
26
(
1
), pp.
49
–
65
.
13.
Karnovsky, I. A., and Lebed, O. I., 2001, Formule for Structural Dynamics: Tables, Graphs and Solutions, McGraw Hill, Inc., pp. 299–304.
14.
Gorman
,
D. J.
,
2000
, “
Free Vibration and Buckling of Inplane Loaded Plates with Rotational Edge Support
,”
J. Sound Vib.
,
229
(
2
), pp.
755
–
773
.
15.
Leissa, A. W., 1969, Vibration of Plates, NASA SP-160.
Copyright © 2002
 by ASME
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