#### Introduction

Renewed interest in filament winding complex composite shell structures has increased the need to examine the current methods to analyze the stress and displacements in such shells. Traditionally, membrane theory, along with netting theory, has been used to calculate stresses for a planar wound composite shall (Hartung, 1963). Through netting theory, optimum dome contours can be developed that closely resemble ellipsoids. These shells are used extensively in the construction of highly efficient pressure vessels for gas storage and rocket motor cases. However, even using the simplified concepts in membrane theory, the complexities of shell theory in conjunction with composite materials limit the types of problems that can be analyzed using non-numerical methods. This paper examines one simple shell structure, the open-ended ellipsoid, and its simplified spheroid shell to investigate the usefulness of membrane theory in filament wound shell structures. Bending stresses are investigated in the open crown region by numerical methods.
Originally Galletly and Radok (1959) examined the effect of bending on an open ellipsoid shell with an open crown, and found that it was small when compared to the in-plane stress resultant. Also, Fettahlioglu and Koenig (1988) developed a solution for a general filament wound shell with an open crown and found the bending stresses small near the free edge. However, their example problem did not take into consideration the changing thickness or material properties and thus would under estimate the bending stresses. Others, such as Nagahdi and De Silva (1955) and Logan and Widera (1989) have presented higher order theories that include bending stresses for ellipsoidal isotropic shells, but the mathematics involved prohibit closed form solutions for all but the simplest conditions, such as constant thickness, non-varying material properties and closed domes. Logan and Hourani (1983) presented results showing the deflections for a laminated orthotropic ellipsoidal shell with constant thickness and constant material properties in the meridian direction, neither of which is valid for a filament wound shell.

This paper uses membrane theory as a basis for deriving the stress relationships and radial displacements for an open filament wound shell. It is found that membrane theory is not sufficiently accurate to predict fiber stresses in the polar region where the shell thickness becomes large and bending begins to occur. Although bending theory has not been included at this time, the results are still valuable for comparison purposes.

© 1997 Kurt Gramoll, The University of Oklahoma, All Rights Reserved.