Stability analysis of thin cylindrical shells under pure and three-point bending

Cylindrical shells curved in only one direction show an interesting behaviour when bent, especially if they remain completely in the elastic region and do not undergo plastic forming. This can be observed in their most common application: the measuring tape. They can be coiled easily because of the loss of stability of their cross-sections, which makes transportation of long shells efficient. This property could be very useful if one could use such a one-way curved shell as a beam, which could be transported and deployed easily. The aim of this study is to observe the behaviour of such a shell, under pure bending load, with special emphasis on the stability loss of the cross-section. In this paper, analytical, semi-analytical, and finite-element methods are used for the description of the shell. The solution derived here uses a variable cross-section Euler-Bernoulli beam model combined with elements of the Kirchhoff plate theory without the shallow shell assumption. It is assumed that the cross-section remains circular and does not change its length. For a universal description, dimensionless parameters and variables are introduced. The semi-analytical investigation revealed that the snap-through ability of the shell may not exist for certain cross-sections which can be presented on a stability map. The derived model reveals the existence of a limiting point between the cross-section deformation modes for larger cross-section angles. In the article, ready-to-use analytical and semi-analytical solutions are given for the critical load and stability map of these shells, which are compared to similar shallow shell models from the literature and the finite-element solution of the problem. The finite-element method also revealed that for a dimensionless description, a length-cross-section radius parameter should be introduced to describe the three-point bending scenario.