Effect of Initial Membrane Stresses on Mode Shapes of Shell Structures

The frequencies and mode shapes of a cylindrical shell were examined in a recent News. In the present News, we demonstrate the effect of initial tensile hoop and longitudinal stresses on these frequencies and mode shapes. The shell geometry, boundary conditions, and material properties are all identical to those considered in the previous News. An initial hoop stress of 20×10⁶ and an initial longitudinal stress of 10×10⁶ are applied to all four cylindrical shells with different thicknesses. These initial stresses closely resemble the stress state in a pressure vessel with an internal applied pressure of p=20×10⁶×t/R.

Table 1 lists the first 20 frequencies and number of longitudinal half waves (N) and number of circumferential full waves (M) for each pre-stressed cylindrical shell. The corresponding results for the unstressed shell (reported in the earlier News) are shown in Table 2 for ease of comparison. The mode shapes corresponding to the lowest frequencies for t = 0.00001 with and without the initial stresses are shown in the movies above. The results warrant some interesting observations:

Table 1  Frequencies and mode shapes of the pre-stressed shell

Table 2  Frequencies and mode shapes of the unstressed shell

• For t = 0.01 there is a small increase in the frequencies but no change in the mode shapes (N and M are unchanged).
• For t = 0.001 there is a larger increase in the frequencies and a minor change in the mode shapes (the 15^th and 16^th frequencies now correspond to M = 4, N = 2 instead of the 17^th and 18^th frequencies in the unstressed shell).
• For t = 0.0001 there is a significant increase in all reported frequencies and also a significant change in the mode shapes. The lowest four frequencies correspond to fewer circumferential waves when compared to the results from the unstressed shell.
• For t = 0.00001 the results are very similar to the results for t = 0.0001, in terms of frequency values and mode shapes. The shell is significantly stiffer than the unstressed shell of the same thickness.
• The asymptotic behavior of the shell (as its thickness is reduced) is different for the unstressed and pre-stressed shells. For the unstressed shell, the lowest frequency scales approximately as √t, while the initially stressed shell results converge to a thickness invariant lowest frequency, that is, the frequency stabilizes at a value above 500 Hz.

The axial modal stresses for the same two cases are given in the Figures (a) and (b) below. The left half in each Figure shows the stress on the cylinder's outer surface and the right half shows the stress on the inner surface (shown side by side in the figure for ease of comparison). For the pre-stressed cylinder there is little difference between the two sides indicating a predominantly membrane deformation mode. For the unstressed case, close to the center of the cylinder there are both bending and membrane deformations, as evident by the difference in the axial modal stress. Near the fixed ends, however, there is little difference between the stresses at the top and bottom surfaces indicating mostly membrane deformations in those regions.

Figure Axial modal stresses for lowest frequency for cylinder
with t = 0.00001   (a) pre-stressed (b) unstressed

This example further illustrates the complex behavior of shell structures, in particular, initial stresses can significantly affect the response of a structure and need to be taken into account to obtain accurate results. As shown above, ADINA can effectively handle the complexities of such analyses.
Keywords:
Shells, frequency analysis with pre-stress effect, mode shapes, initial stress, membrane-bending coupling