On the Torsion of Elastic Cylindrical Bars

Certain trigonometric formulae and combinatorial identities are used to derive exact and closed expressions for the stress functions, torsional rigidities and peripheral shearing stresses of certain isotropic cylinders under torsion. The cross sections are bounded by the closed curves \documentclass{article}\pagestyle{empty}\begin{document}$ r = a\left| {\sin \frac{\theta }{n}} \right|^n \left({ - \pi < \theta \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} \pi } \right) $\end{document} where n is a positive integer (n > 1). Complex variable methods are applied to obtain the complex torsion functions, torsional rigidities and shearing stresses when the sections are bounded by the contours \documentclass{article}\pagestyle{empty}\begin{document}$ r = a\cos ^n \frac{\theta }{n}\left({ - \frac{{n\pi }}{2} < \theta \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} \frac{{n\pi }}{2}} \right) $\end{document} , where n is a positive number (0 < n ≦ 2). Numerical results are presented in the form of tables and graphs illustrating the variation of torsional rigidities and the distribution of tangential shearing stresses along the edges of the cross sections. The cardioid and lemniscate of Bernoulli are included as special cases.