It is shown that the utilization of the Filonenko-Borodich set of functions, as the comparison functions in the Boobnov-Galerkin method leads to the result that coincides with the closed-form solution for the clamped-clamped uniform beam under uniformly distributed load. It is hoped that this remarkable, direct coincidence could be used in graduate courses and books on mechanics of solids.
Issue Section:
Brief Notes
1.
Galerkin
, B. G.
, 1915, “Beams and Plates,” Vestnik Inzhenerov, 1, pp. 897–908, (see also Collected Works, USSR Academy of Sciences Publishing, 1952, pp. 182–183).2.
Duncan, W. J., 1937, “Galerkin’s Method in Mechanics and Differential Equations,” Aeronautical Research Council Reports and Memoranda No. 1798.
3.
Duncan, W. J., 1938, “The Principles of the Galerkin Method,” Aeronautical Research Report and Memoranda No. 1894.
4.
Mikhlin, S. G., 1964, Variational Methods in Mathematical Physics, Pergamon Press, Oxford.
5.
Hoff, N. J., 1956, The Analysis of Structures, Wiley, New York.
6.
Kantorovich, L. V., and Krylov, V. I., 1958, Approximate Method for Higher Analysis, Interscience (English translation), New York.
7.
Ishlinski, A. Ju. (Editor), 1975, Advances of Mechanics of Deformable Continua (dedicated to the 100th Anniversary of the Birth of B. G. Galerkin), Nauka, Moscow (in Russian).
8.
Strang, G., and Fix, G. J., 1973, An Analysis of the Finite Element Method, Prentice-Hall Englewood Cliffs, New Jersey.
9.
Leipholz
, H. H. E.
, 1976
, “Use of Galerkin’s Method for Vibration Problems
,” Shock Vib. Dig.
, 8
, pp. 3
–18
.10.
Singer
, J.
, 1962
, “On the Equivalence of Galerkin’s and Rayleigh Ritz Methods
,” Journal of the Royal Aeronautical Society
,66
, p. 592
592
.11.
Elishakoff
, I.
, and Lee
, L. H. N.
, 1986
, “On Equivalence of the Galerkin and Fourier Series Methods for one Class of Problems
,” J. Sound Vib.
, 109
, pp. 174
–177
.12.
Filonenenko-Borodich
, M. M.
, 1946
, “On a Certain System of Functions and its Applications in the Theory of Elasticity
,” Prikl. Mat. Mekh.
, 10
, pp. 193
–208
(in Russian).13.
Stepanov, V. V., 1945, “On Some Complete Non-Orthogonal Systems,” Proceedings of the USSR Academy of Sciences, Vol. XL VIII, No. 6 (in Russian).
14.
Gradshteyn, I. S., and Ryzhik I. M., 1980, Tables of Integrals Series and Products, Academic Press, New York.
15.
Vilenkin
, N. Ya.
, 1952, “On Some Nearly Periodic Systems of Functions,” PMM, 16(3), pp. 812–814 (in Russian).Copyright © 2003
by ASME
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