GT Lab Delta Pic

Finite Element Analysis

The obvious question now is how accurate is membrane theory for open elliptical filament wound shells? The best method to verify a solution is, of course, by experimental methods, but this is not always possible due to physical constraints of experimental equipment and costs. Another way to investigate the accuracy of membrane theory is to use higher order solutions that include bending and shear effects. However, for even the simpler homogeneous elliptical shell (Galletly and Radok, 1959, and Nagahdi and De Silva, 1955) or the filament wound spherical shell (Fettahlioglu, 1969), it is extremely difficult mathematically to develop a closed form solution. A third way is to use numerical methods, such as finite elements (FE), to obtain a solution which is used in this paper.

To investigate the stresses and displacements of open-ended composite shells using finite elements, two types of elements are commonly available, axisymmetric and shell. Both can be formulated to accept anisotropic materials such as fiber reinforced plastics used in filament winding. Of the two elements, the axisymmetric element is the more common for composite shells of revolution, mainly due to its simplicity and wide availability. The layered composite shell element is a more recently developed element and has more capabilities to model filament wound layered structures. However, for rapidly changing thickness, like near the dome opening, shell elements have a difficult time modeling the bending stresses accurately. For this study, only geometric nonlinear axisymmetry elements were used.

As shown in Figs. 8-11, the displacements for a composite shell are much larger than for a homogeneous shell of comparable strength, suggesting the need for a large displacements nonlinear FE analysis. Linear FE assumes small displacements and rotations plus infinitesimal strains, whereas geometric nonlinear FE analyses assume the geometry will undergo large displacements and rotations, and finite strains (material nonlinearity is also possible, but is not considered in this study). All FE analyses performed in this paper were done using the ANSYS finite element program commercially available from Swanson Analysis Systems, Inc.

For comparison with the membrane solution for stresses and displacements, a FE analysis was performed for each of the three shells. The FE analysis used 4-node qualidrilateral isoparametric, axisymmetric elements. The material properties varied along the meridian as required by the filament winding process. Due to the thinness of the shell, only three elements were used through the thickness with aspect ratios of up to 3. The total number of nodes in the model were approximately 3,500. The fiber direction stress results for an internal pressure load of 1 MPa are shown in Fig. 12-14.

In both the ellipsoid and spheroid shells, the membrane theory under-predicts the stresses, particularly near the dome opening. This is not surprising since bending stresses are not included in membrane theory. However, it is interesting to note that membrane theory does not even predict stresses away from the boundaries, even though this is a very thin shell. In particular, both the ellipsoid and spheroid shows a bending stress throughout the complete shell. The netting theory does not match either FE or membrane theory results since netting theory is really only good for netting dome shells. The FE results also demonstrate that geometric nonlinear conditions should be taken into account when performing a numerical analysis of a composite shell.

The shear and transverse stresses shown in Figs. 5-7 vary between the FE results and membrane theory predictions, particularly away from the dome opening. For the ellipsoid and spheroid shell, knowing the correct shear and transverse stress is important, since they are likely to failure in those modes.

The radial displacements predicted by FE are lower than those predicted by membrane theory, which is expected since membrane theory is a linear theory and the FE analysis allows for large deformation. The large deformation allows displacement adjustments as the shell is pressurized, which, in turn, causes the shell to deflect more uniformly. This is worst for the ellipsoid shell because of the compression stress that changes with pressurization. The spheroid will undergo less deformation due to its spherical shape that remains fairly constant with pressurization.

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