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.

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.