Abstract
The intuitive ideas underlying St. Venant’s principle [1] in the mathematical theory of elasticity are well known. The principle, stated roughly, asserts that if self-equilibrated loads are applied on one end only of a long cylinder, then the strain produced in the body by such forces are much larger near the loaded end than at points far from the loaded end. The principle is now over one hundred years old, yet it remains a vaguely stated intuitive notion without a precise mathematical formulation capable of proof. The theorems of Boussinesq [2], von Mises [3], and Sternberg [4] are often mentioned in connection with St. Venant’s principle, but I have been unable to perceive an easy relationship. Dou [5] proves yet another property of the solutions of elasticity theory for isotropic cylinders having a square cross-section which I also find difficult to translate into St. Venant’s principle. The results of Zanaboni [6, 7] are in the right direction but are not sufficiently sharp to be considered definitive.
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References
de Saint-Venant, B.: Mem. Acad. Sci. savants étrangers 14 (1855) 233–560, and J. Liouville, s. 2, 1 (1856) 89-189.
Boussinesq, M. J.: Applications des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques, Paris: Gauthier-Villars 1885.
von Mises, R.: On Saint-Venant’s Principle. Bull. Amer. Math. Soc. 51 (1945) 555–562.
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Dou, A.: On the Principle of Saint-Venant, MRC Technical Summary Report No. 472, Mathematics Research Center, The University of Wisconsin, Madison, Wisconsin, May 1964.
Zanaboni, O.: Dimostrazione generale del principio del De Saint-Venant. Atti Acc. Naz. Lincei 25 (1937) 117–120.
Zanaboni, O.: Valutazione dell’errore massimo cui dà luogo l’applicazione del principio del De Saint-Venant. Atti Acc. Naz. Lincei 25 (1937) 595–601.
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© 1966 Springer-Verlag Berlin Heidelberg
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Toupin, R.A. (1966). On St. Venant’s principle. In: Görtler, H. (eds) Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-29364-5_15
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