Open AccessThe flexure and torsion of an internally cracked shaft
Author(s) -
L. A. Wigglesworth
Publication year - 1939
Publication title -
proceedings of the royal society of london. series a, mathematical and physical sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.814
H-Index - 135
eISSN - 2053-9169
pISSN - 0080-4630
DOI - 10.1098/rspa.1939.0037
Subject(s) - rectangle , torsion (gastropod) , rigidity (electromagnetism) , elliptic integral , mathematics , bounded function , mathematical analysis , geometry , coordinate system , structural engineering , engineering , medicine , surgery
In § 2 of this paper an integral formula is derived for a potential function existing in a two-dimensional regionS bounded by two curvesC 1 andC 2 , and having assigned values on these boundaries. The formula can be applied if the potential function existing in S and having constant values on the boundaries is known, or in other words, when the area S, made simply connected by a suitable cross-cut, can be represented conformally on to the interior of a rectangle by a known transformation. These results are later applied to the problem of the flexure and torsion of a shaft whose cross-section is a circle with an interior slit, the slit being a segment of a diameter, not necessarily in the central position. With the aid of the results of § 2 and the appropriate transformation the problem is solved in finite terms of elliptic functions, and the torsional rigidity and centre of flexure are given also in finite terms. In two cases, when the slit is central, and when it starts from the centre, the torsional rigidity and the position of the centre of flexure (in the second case) are given in terms of complete elliptic integrals, and numerical results have been worked out for slits of different lengths.