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Torsion of beams of ⊥- and ∟- cross-sections

Published online by Cambridge University Press:  24 October 2008

Extract

The problem of the torsion of beams of - and L-cross-sections has received attention from very few authors despite its important technical applications. The first mathematical solution in this connection was obtained by F. Kötter in 1908 for an L-section both of whose arms are infinite. He attacked the problem by the use of the known solution of the rectangle and by application of the scheme of conformal transformation. Kötter's method, however, does not lend itself readily to the solution of the problem involving more than one re-entrant angle. The first solution for the torsion of a beam whose cross-section is a rectilinear polygon of n sides was published in 1921 by E. Trefftz who also applied his method to an infinite L-section. Recently I. S. Sokolnikoff has suggested a more general method depending upon the fundamental theorem of potential theory that a harmonic function is uniquely determined by the values assigned along the boundary of the region within which the harmonic function is sought, the boundary condition and the region being subject to certain well-known assumptions of continuity, connectivity, etc. As an illustration of his method he has given an approximate solution for a -section whose flange and web are both infinite.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

* Kötter, F., Sitzungsberichte der Preuss. Akad. der Wiss. (1908), 935955.Google Scholar

Trefftz, E., Math. Ann. 82 (1921), 97112.CrossRefGoogle Scholar

Sokolnikoff, I. S., Trans. American Math. Soc. 33 (1931), 719732.Google Scholar

* In this formula it is assumed that the constant in the boundary value of ψ given by (4) is zero.

See p. 104 of his paper quoted on p. 392.

* For a general analytical proof in this connection see Todhunter and Pearson, History of Elasticity, Vol.2, Part 2, 412414.Google Scholar

* Cf. Greenhill, , Applications of elliptic functions, 286.Google Scholar

* See pp. 730–731 of his paper quoted on p. 392.