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Series solution of the biharmonic equation in the rectangular domain with some applications to mechanics

Published online by Cambridge University Press:  24 October 2008

H. Vaughan
H. Vaughan
Affiliation:
University of British Columbia, Vancouver 8, Canada
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Summary

A double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.

The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.

In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 3 , November 1974 , pp. 563 - 585
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Benthem, J. P.Quart. J. Mech. Appl. Math. 24 (1963), 115.Google Scholar
(2)Buchwald, V. T.Proc. Roy. Soc. Ser. A, 277 (1964), 385.Google Scholar
(3)Fadle, J.Ing. Arch. 11 (1941), 125.CrossRefGoogle Scholar
(4)Faxèn, O. H.Z. Angew. Math. Mech. 15 (1935), 268.CrossRefGoogle Scholar
(5)Gaydon, F. A. & Shepherd, W. M.Proc. Roy. Soc. Ser. A, 281 (1964), 184.Google Scholar
(6)Gaydon, F. A.Proc. Roy. Soc. Ser. A, 283 (1965), 356.Google Scholar
(7)Gaydon, F. A.Proc. Roy. Soc. Ser. A, 286 (1965), 251.Google Scholar
(8)Green, A. E.Proc. Cambridge Philos. Soc. 40 (1944), 222.CrossRefGoogle Scholar
(9)Green, A. E. & Zerna, W.Theoretical elasticity, 2nd ed. (Oxford University Press, 1968).Google Scholar
(10)Horvay, G.J. Appl. Mech. 20 (1953), 87.CrossRefGoogle Scholar
(11)Johnson, M. W. & Little, R. W.Quart. Appl. Math. 22 (1965), 335.CrossRefGoogle Scholar
(12)Kato, H.J. Japanese Soc. Nav. Arch. 50 (1932), 209.Google Scholar
(13)Lévy, S.J. Appl. Mech. 9 (1942), 171.CrossRefGoogle Scholar
(14)Love, A. E. H.Proc. London Math. Soc. 29 (1929), 189.CrossRefGoogle Scholar
(15)Morley, L. S. D.Quart. J. Mech. Appl. Math. 16 (1963), 109.CrossRefGoogle Scholar
(16)Semple, H. M. & Alford, M. J.Quart. J. Mech. Appl. Math. 23 (1970), 209.CrossRefGoogle Scholar
(17)Semple, H. M. & Alford, M. J.Quart. J. Mech. Appl. Math. 24 (1971), 115.CrossRefGoogle Scholar
(18)Sezawa, K.Rep. Aero. Res. Inst. Tokyo Imp. Univ. no. 69 (1931).Google Scholar
(19)Sezawa, K.Rep. Aero. Res. Inst. Tokyo Imp. Univ. no. 70 (1931).Google Scholar
(20)Sezawa, K. & Watanabe, W.Rep. Aero. Res. Inst. Tokyo Imp. Univ. no. 143 (1936).Google Scholar
(21)Smth, R. C. T.Austral. J. Scientific Research 5 (1952), 227.Google Scholar
(22)Sneddon, I. N.Fourier transforms (McGraw-Hill, 1951).Google Scholar
(23)Taylor, G. I.Z. Angew. Math. Mech. 13 (1933), 147.CrossRefGoogle Scholar
(24)Tiffen, R. & Semple, H. M.Mathematika 12 (1965), 193.CrossRefGoogle Scholar
(25)Timoshenko, S. P. and Goodier, J. N.Theory of elasticity, 3rd ed. (McGraw-Hill, 1970).Google Scholar
(26)Timoshenko, S. P. and Woinowsky-Krieger, S.Theory of plates and shells, 2nd ed. (McGraw-Hill, 1959).Google Scholar
(27)Tomotika, S.Philos. Mag. 21 (1936), 745.CrossRefGoogle Scholar
(28)Warburton, G. B.Proc. Inst. Mech. Engrs. 168 (1954), 371.CrossRefGoogle Scholar
(29)Weinstein, A.J. London Math. Soc. 10 (1935), 184.CrossRefGoogle Scholar
(30)Weinstein, A.Proc. Cambridge Philos. Soc. 32 (1936), 96.CrossRefGoogle Scholar
(31)Young, D.J. Appl. Mech. 17 (1950), 448.CrossRefGoogle Scholar