Sage Journals: Discover world-class research

Abstract

In dell' Isola and Ruta, and dell' Isola and Rosa is suggested a "perturbative approach" (Nayfeh; Trabucho and Viaho) to the Saint-Venant problem for thin cross sections, however, the papers deal with closed cross sections or with cross sections of constant thickness only (see also Wheeler and Horgan). Here we generalize the proposed procedure by giving a method for treating the case of open or closed sections of variable thickness. We find all the known formulas (Trabucho and Viafio; Feodosyev; Chase and Chilver; Baldacci) due to Kelvin and Bredt as first non-vanishing-terms of our perturbative development and give the corrections to these formulas, too. This seems to be a first step toward solving the open problem formulated in Trabucho and Viafio, pp. 162-164.

Get full access to this article

View all access options for this article.

References

[1] dell' Isola, F. and Ruta, G. : Outlook in Saint-Venant Theory I: Formal expansions for torsion of Bredt-like section. Arch. Mech., 46, 6, 1005-1005 (1994).

[2] dell' Isola, F. and Rosa, L. (in press): Perturbation methods in torsion of thin hollow Saint-Venant cylinders. Mechanics Research Communications.

[3] Nayfeh, A. : Perturbation Methods, John Wiley and Sons, New York, 1973.

[4] Trabucho, L. and Viailo, J. M. : Mathematical Modelling of Rods, Elsevier, North Holland, 1995.

[5] Wheeler, L. and Horgan, C. O. : Isoperimetric bounds on the lowest nonzero Stekloff eigenvalue for plane strip domains, Siam J. Appl. Math, 31, n. 2 (1976).

[6] Feodosyev, V.: Soprotivlenie Materialov, MIR, Moskwa, 1968. [Italian translation: Resistenza dei materiali, Editori Riuniti, Roma, 1977].

[7] Chase, J. and Chilver, A. H. : Strength of Materials and Structures, Edward Arnold, London, 1971.

[8] Baldacci, R.: Scienza delle costruzioni, UTET, Torino, 1970.

[9] Bredt, R.: Kritische Bemerkungen zur Drehungselastizitfit, Zeits. Ver. deutsch. Ing. 40, 815 (1896).

10.

[10] Prandtl, L.: Zur torsion von prismatischen Stiben, Phys. Zeits. 4, 758 (1903).

11.

[11] Fraeijs de Veubeke, B. M. : A Course in Elasticity, Springer-Verlag, New York, 1979.

12.

[12] Clebsh, A. : Thiorie de l'6lasticiti des corps solides (Traduite par MM. Barre de Saint-Venant et Flamant, avec des Notes 6tendues de M. Barre de Saint-Venant), Dunod, Paris, 1983 [Reprinted by Johnson Reprint Corporation, New York, 1996].

13.

[13] Sokolnikoff, I. S. : Mathematical Theory of Elasticity, McGraw-Hill, New York, 1946.

14.

[14] Love, A.E.H. : A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 1949.

15.

[15] Axler, S. , Bourdon, P. and Ramey, W. : The Harmonic Function Theory, Springer-Verlag, New York, 1992.

16.

[16] Weatherburn, C. E. : An Introduction to Riemmanian Geometry and the Tensor Calculus, Cambridge University Press, MA, 1963.

17.

[17] Germain, P.: Cours de mecanique des milieux continus, Tome 1,2, Masson, Paris, 1973.

18.

[18] Kobayashi, S. and Nomizu, K. : Foundations of Differential Geometry, Vol. 1,2, Interscience, New York, 1969.

19.

[19] Caratheodory, C. : Theory of Functions of a Complex Variable, Vol. 1,2, Chelsea, New York, 1954.

20.

[20] Timoshenko, S. and Goodier, J. N. : Theory of Elasticity. McGraw-Hill, New York, 1951.

21.

[21] Wheeler, L. and Horgan, C. O. : Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure. J. of Elasticity, 6, n.4 (1976).

An Extension of Kelvin and Bredt Formulas

Abstract

Get full access to this article

References