[Univ of Cambridge] [Dept of Engineering]
Equilibrium and stability of inflatable membrane structures

B. Andrew Lennon


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.

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