Nonlinear Bending of Straight or Initially Curved Pressurised Thin-Walled Sections
van Steen, R.
Delft University of Technology

Pure bending of thin walled sections has been subject of study ever since 1927. It was in this year Brazier derived a first estimate on the nonlinear static bending response of a gradually ovalising circular cross-section using a simple variational approach. This type of nonlinear bending progressing into a flattened and finally collapsing section has become known as ``Brazier bending'' or ``limit load bending''. Wood and Kedward added pressure loading and material orthotrophy to Braziers' model respectively. Reissner firstly derived equilibrium equations using a theory of finite bending of circular rings. A plethora of models and numerical solutions procedures basically all solving Reissners equations or close variations thereoff is what followed. Using small well selected additions to the kinematics of the problem, Tatting derived a particularly accurate approximation of static response. Next to the addition pressure and orthotrophy on initially straigth tubes, curved tubes also have been considered since decades. Von K\'arm\'an already studied curved tubes earlier than Brazier published his renowned 1927 paper. Without summarizing any further literature on the bending of sections the following can be stated about most of the models. All use simplified kinematics such as the assumption of an inextensional circumference or moderate rotation. Unknown ranges of validity are troublesome when going into the analysis of inflatables. It is the inflatable type of thin walled structures asking for new models with extended range of validity and modelling options.

However, a more generally valid model (large strain and rotations kinematics, varying pressure along with deformation, pre-stress in a straight section, initially curved and or non-circular sections) generally is hard to obtain. When aiming for an analytical model or even a series solutions equations quickly become to elaborate or unwanted linearisations are to be made. When going into full (3- dimensional) finite element models difficulties arise in the sense that boundary conditions inevitably have to be dealt with.

A model for a section showing large strain and large rotation along with capability of describing pressure and material is a good starting point in studying inflatable structures. Inflated structural members typically are pressurised closed section showing large deformation.

This contribution presents a 2-dimensional finite element model of co-rotating beam elements with a special orthotropic set able to describe pure bending of non-straight, non-circular, pressurised sections using a gas law which even is able to represent an incompressible fluid. Not only can this model serve as an analysis tool of pressurised sections, with future additions stability of inflated member can be studied. Even wave length estimates of localised instability (wrinkling) in future could help in building adequate finite element models for more general inflated structures.