A lobed parachute with meridional cords adopts a shape where the cords
take the shape of the isotensoid curve. The isotensoid is the three
dimensional surface that has zero hoop stress and carries load only in the
meridional direction.

This dissertation investigates the equilibrium and stability of inflated
membrane structures. Structures that adopt the shape and the isotensoid
curve are of particular interest. It is shown that the isotensoid has
two forms, the existing axisymmetric form and a polygonal form that is introduced
in this dissertation. It is shown analytically that both forms are
in equilibrium and these results are supported by finite element analyses.
The inflated shape of these structures depends on stability considerations
in addition to the equilibrium requirement. The potential energy for
inflated structures is proportional to the negative of the enclosed volume
hence the configuration with the largest enclosed volume will be the most
stable configuration.

Two example problems show the importance of stability considerations in
inflated membranes. The first example concerns the stability of a
membrane structure, such as a “pumpkin” balloon, with longitudinal lobes
separated by load bearing meridional cords in the shape of isotensoid curves.
A deformation mode is postulated and calculations show that structures in
this mode can have a larger enclosed volume than structures in the nominal
configuration. The relationship between the volumes in the normal
and deformed configurations depends on the lobe size and the number of lobes.
The volume initially decreases with increasing perturbation size, showing
that the nominal configuration is stable for an infinitesimal perturbation.
However, for large numbers of lobes only a small perturbation is required
for the deformed configuration to be more stable than the nominal configuration.

The second example concerns the inflated shape of a structure made from
two identical, circular membrane sheets, joined along their edges.
The final shape of the structure approximates a polygonal isotensoid in the
region away from the apex, with excess material forming folds which separate
the sides of the isotensoid. It is shown that the polygonal isotensoid
with the minimum number of sides encloses the maximum volume hence is the
most stable. This surface requires more material than an axisymmetric
isotensoid, which is provided by the membrane pulling inwards during inflation.
Elastic stretching near the apex limits the size of the polygonal isotensoid
and a geometric constraint at the transition to uniaxial stress can be used
to impose a limit on the minimum number of sides of the isotensoid, hence
determining the number of folds.

Finite element analyses are used to support the work on the shape of the
inflated membrane. These analyses confirm the expected shape and the
stress distribution in the inflated membrane. Additional analyses are
used to study the radius of the transition which is used in conjunction with
the method for predicting the number of folds.

[Cambridge University | CUED | Structures Group | Geotechnical Group ]