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Dynamic Analysis of a Cylindrical Shell
In this technical brief we highlight some of the issues involved in performing an accurate dynamic finite element analysis. A simple model involving a cylindrical shell fixed at both ends is investigated (see Figure below). For this shell structure, the mode shapes corresponding to the lowest frequencies depend significantly on the thickness of the shell. As the shell thickness is reduced, the number of oscillations around the circumference increases. This phenomenon is quite specific to shells and is, for example, not seen in the dynamics of beams.
Figure 1: Cylindrical shell model; both ends are fixed; thickness t
A region of the shell is subjected to a uniform time-varying pressure load. Three values of thickness are considered (t = 0.01, 0.001, 0.0001) to cover typical shell thicknesses. An implicit transient dynamic analysis shall be performed. Hence the first objective is to find an appropriate mesh for the analyses. Three different mesh densities that might typically be used in engineering practice are investigated, all using the MITC4 shell element. The number of elements along the circumferential and longitudinal directions are 40×20 for the coarse mesh, 80×40 for the intermediate mesh and 240×120 for the fine mesh.
The appropriate mesh for a dynamic analysis should be able to accurately capture the static response of the model, and the highest natural frequency content of the dynamic excitation [1]. A mesh that is suitable for a static analysis may therefore not be suitable for dynamics. Also, a mesh suitable for a certain shell thickness may not be suitable for a thinner shell of otherwise identical geometry. Both concepts will be demonstrated in this example.
The errors for the maximum displacements in the static analyses are shown in Table 1. All reference solutions for the errors have been obtained by solving the problems with an extremely fine mesh. By setting an upper error limit of 5%, it is clear that the coarse mesh gives acceptable accuracy for t = 0.01 and t = 0.001, and the fine mesh is required for t = 0.0001.
Dynamic analyses using implicit direct time integration are then performed with an excitation frequency ωL = 2 × ωmin, where ωmin is the lowest natural frequency (different for each shell thickness), and 2% Rayleigh damping around ωL. In each solution, a very small time step is used for the time integration. The errors in the maximum displacements (now also over time) are shown in Table 2. The errors, in most cases, have increased compared to the errors in the static analysis. The coarse mesh is still acceptable for t = 0.01. The intermediate mesh is acceptable for t = 0.001, while for t = 0.0001 the fine mesh is needed to satisfy the displacement error.
The animation above shows a part of the dynamic response for all three shell thicknesses. We see that for the shell with t = 0.0001, the number of oscillations circumferentially is significantly larger than for the shell with t = 0.01.
Table 1 Errors in maximum displacements in static analysis (in percentage)
| t = 0.01 | t = 0.001 | t = 0.0001 | |
| Coarse mesh | 2.57 | 2.51 | 23.5 |
| Intermediate mesh | 0.800 | 0.785 | 7.10 |
| Fine mesh | 0.0370 | 0.0719 | 0.0864 |
Table 2 Errors in maximum displacements in dynamic analysis (in percentage)
| t = 0.01 | t = 0.001 | t = 0.0001 | |
| Coarse mesh | 1.49 | 8.73 | 31.5 |
| Intermediate mesh | 1.11 | 0.353 | 10.1 |
| Fine mesh | 0.170 | 1.01 | 3.01 |
ADINA also has other options for dynamic analysis, namely mode superposition, frequency domain analyses, and explicit time integration. In dynamic analyses, conservatively, an accurate mesh should capture all natural frequencies up to 4 × ωL [1]. Explicit dynamic analysis is suitable for cases where the excitation includes high frequencies, as in wave propagation problems, and the stability limit then usually governs the time step to be used.
For this shell problem, a transient implicit dynamic analysis provides an efficient solution.
This example highlights some of the issues that the user must consider when selecting an appropriate analysis type and mesh for a dynamic problem. In all cases, it is important to use a reliable and powerful code like ADINA, with all the above-mentioned various options available for an efficient dynamic analysis.
References
Keywords:
Dynamics, shells, frequencies, transient analysis, mesh convergence, modeling, MITC4 shell elements