# Modelling the Equilibrium and Stability of Slit Tubes

**by Diana Galletly**

*Abstract*:

A novel shell structure, a bi-stable composite slit tube, is presented.
This structure is formed like a coilable tape measure; unlike tape measures,
however, it is stable in both the extended form and the coiled form, without
any need for a spindle or casing to hold it in either position.

The structure is initially modelled linearly, as a beam. Various
applications of this model are presented: an algorithm for rapidly locating
the second equilibrium point, the investigation of the strain state as the
path is followed from one equilibrium state to the second, and generalised
plots throughout longitudinal curvature--transverse curvature space.
Various examples (two composites with layups which are anti-symmetric about
the middle layer, one which is symmetric about the middle layer, and an
isotropic case) are presented. The inability of this model to determine
the stability of an equilibrium is noted, as the results for the anti-symmetric
and symmetric cases appear to be similar, whilst their experimental behaviour
is very different. The model is then extended to incorporate non-linear geometric
effects, and stability criteria are determined. The examples considered
for the original model are re-evaluated using the new model; the equilibrium
locations for the anti-symmetric and isotropic cases are found to be unchanged,
whilst the

symmetric case develops more twist. The stability of the anti-symmetric
layups, and the instability of the symmetric layup and the isotropic case,
are accurately predicted.

The structure is then modelled as a shell of arbitrary cross-section.
An expression for this cross-sectional shape is derived, and the same examples
once again examined. The stability of the anti-symmetric layups, and
the instability of the symmetric and isotropic cases, are again predicted
correctly. The constant transverse curvature assumption of the extended
beam model is found to be valid for cross-sectional angles greater than
about 180 degrees.

Various reasons for the disparity between the experimental and theoretical
results are then investigated, including inaccuracies in the material properties
and inelasticity in the matrix. It is thought that matrix inelasticity
is the cause of the difference and initial results to support this are presented.

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(last update 10 November 2003)