Table 1.4.1–2 summarizes the results obtained using the axisymmetric shell elements SAX1 and SAX2 for the first 10 modes. For the lower-order modes and the thinner shell case, the results agree well with membrane theory. The natural frequency of the ninth mode for  0.05 is significantly different from that predicted by membrane theory and is in agreement with Kalnins (1964). Membrane theory is clearly accurate for small values of and for the lower-order modes. The mode 1 corresponds to rigid body translation and is not shown in the table. In the axisymmetric case each frequency has a distinct mode shape and the eigenvalue iterations converge rapidly.

Table 1.4.1–3 and Table 1.4.1–4 summarize the results obtained using the asymmetric-axisymmetric shell elements SAXA1N and SAXA2N (N=1, 2, 3 or 4). In this case for each value of there are +1 modes instead of +1, as predicted analytically. This is because, in the asymmetric-axisymmetric element formulation, symmetry with respect to the – plane at 0 is assumed. However, for each the number of modes computed is limited by +1, where is the number of Fourier interpolation terms used.

Recall that, in the full models using general shell elements, there are +1 modes for each value of . To improve convergence in the eigenvalue iteration, we have, therefore, specified a higher number of trial vectors to be used. We calculate 18 eigenvalues to get the modes up to 3. For higher-order modes such as 9, at least 100 or more eigenvalues need to be calculated. To keep this qualification test within a reasonable computational time, we have restricted the number of eigenvalues to 20. It implies that the bending effects will not be visible to the same extent as in the axisymmetric case. For this reason results from the general shell models are reported here only for the thin shell case with  0.01.

Table 1.4.1–5 provides the results for second-order shell elements; Table 1.4.1–6 provides the results for first-order shell elements. In these tables we list the first 20 eigenvalues, except the first six rigid body modes.

When second-order shell elements are used, the first five values (7 through 11) are almost identical to the membrane solution for the 2 case. The first-order mesh uses the same number of elements as the second-order mesh. Nevertheless, except for S3R elements, the results are quite accurate: the error is less than 2% for the first five eigenvalues. For S3R elements the maximum error is around 5% because these elements use a constant bending strain approximation. The accuracy can be increased by further refining the mesh. Eigenvalues 12 through 18 correspond to the mode 3. It is observed that +1 modes are recovered, as predicted by the analytical solutions.

We also notice that the first-order triangular elements show more variance in eigenvalues corresponding to a given value of than the quadrilaterals. This is a consequence of orientation effects of the triangular element. The accuracy could be improved by designing the mesh to be spherically symmetric.

Figure 1.4.1–2 illustrates the modes 2 and 3 obtained with any of the shell models used.