On the Torsion of the Slit Circular Cylindrical Shell

The complexity of shell analysis stems from the fact that loads are resisted, in general, by both membrane and flexural action. There is a need to develop a suitable shell finite element, but many of those proposed at the present time fail in the context of the “sensitive solutions” coupled with rigid body movements. These “sensitive solutions” refer to known solutions (within the framework of shell theory) of problem in which it is kinematically possible for the shell to deform with no straining of the midsurface. (Purely inextensional solutions were first considered by Lord Rayleigh [1].) In two cases, namely the torsion of a slit cylinder and the application of uniformly distributed moments M_θ and M_x = vM_θ to a slit cylinder, the contribution of the membrane forces is identically zero. The second of these two cases is too trivial to be used for comparison with finite element solutions, but the first case exhibits many of the features that current finite elements have difficulty in reproducing. An exact solution within the context of shell theory is not necessarily, of course, an exact three-dimensional solution. However, the torsion of a slit cylinder has also been solved three-dimensionally (Love [2]) and the present Note focuses attention on the exact variation of shear stress through the wall of the shell and on how this compares with shell theory.