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Torsion of beams whose cross-section is a regular polygon of n sides

Published online by Cambridge University Press:  24 October 2008

B. R. Seth
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Extract

E. Trefftz has discussed the problem of the torsion of a beam whose cross-section is bounded by a polygon with the help of the Schwarz-Christoffel transformation given by

where a1, a2, …, an are external angles of the polygon in the w-plane, and ξ1, ξ2, …, ξn are the points on the real ξ-axis in the t-plane that correspond to the angular points of the polygon in the w-plane. In the case of regular polygons a further transformation of the upper half of the t-plane into the interior of a circle in the z-plane with the help of the transformation

greatly simplifies the problem, and some definite results can be obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 2 , 30 April 1934 , pp. 139 - 149
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

* Math. Ann. 82 (1921), 306–319.Google Scholar

* This result is due to Schwarz. See Forsyth, , Theory of functions, 3rd edition, p. 768, Ex. 2.Google Scholar

* Cf. Dixon, A. C., Quart. Journ. of Math. 24 (1890), 167–233.Google Scholar

* Richelot, , Journal für Math. 9 (1832), 407–408Google Scholar, also Cayley, , Proc. Camb. Phil. Soc. 4 (1881), 106–109.Google Scholar

* Thomson, and Tait, , Natural Philosophy, Vol. i, Part 2, pp. 249–250.Google Scholar

* That this supposition of Boussinesq is not always true has been shown by Filon, L. N. G., Phil. Trans. Roy. Soc. 193 (1900), 309–352.CrossRefGoogle Scholar