In the preceding chapters on birods, beamshells, and axishells, we studied three important classes of structures that, under special loads and boundary/initial conditions, suffer strains, rotations, and displacements that are functions of one spatial variable only. The resulting simplifications of the general governing equations are enormous and in static problems, partial differential equations reduce to ordinary ones. Because strains, unlike rotations or displacements, are unaffected by rigid-body motions, it seems natural and possibly fruitful to seek other shells that, subject to suitable loads and boundary/initial conditions, suffer deformations in which the strains depend on one spatial variable only but in which the rotations and displacements depend (possibly) on a second one. The simplest and most important example is the bending of a pressurized tube of circular planform and arbitrary cross section—the subject of Sections A and B.

Specifically, in Section A we consider tubes subjected to a net end moment and, if pressurized, to an equilibrating net end force as well. We formulate the governing equations, discuss various approximations, and survey the literature on the collapse of such tubes. In Section B, we develop variational principles for curved tubes of either closed or open cross section, allowing for loads that act on the boundaries of the latter.

Beginning with Section C, we extend our analysis to unishells, i.e., to any shell with a set of loads and boundary/initial conditions (which must include the trivial ones that produce no distortion) such that there is a system of coordinates (σ, τ) on the reference surface in which the components of the deformed metric and curvature tensors are independent of τ.