Surprises in Mode Shapes of Shell Structures
The dynamic analysis of shell structures is a most interesting subject. Physical phenomena can be discovered that, at first sight, may appear to be counterintuitive.
We reported earlier on the solution of frequencies and mode shapes of a shell structure with contact, and on the dynamic analysis of a cylindrical shell. We now want to consider the cylindrical shell of the dynamic analysis and give some interesting — and perhaps surprising — results regarding the frequencies and mode shapes of the structure as its thickness is decreased.
Figure 1 shows the shell structure considered. We solve for the lowest 20 frequencies and mode shapes for different shell thicknesses, and consider t = 0.01, 0.001, 0.0001, and 0.00001.
Figure 1 Cylindrical shell model; both ends are fixed; thickness t
Table 1 Frequencies of the shell as t decreases
Table 1 lists the frequencies and the number of longitudinal "half" waves (N) and number of circumferential full waves (M) in each mode. The name "half" wave is used to describe the shape, although the slope is zero at the ends. We can make the following interesting and sometimes surprising observations.
The number of circumferential full waves is almost always larger than the number of longitudinal half waves. Only when t = 0.01 (the moderately thick shell case) do we have that for some mode shapes N = M, and only for the 18th and 19th frequencies, do we have that the number of full longitudinal and circumferential waves is the same, here N = 2, M = 1, see the movie in Figure 2.
f18 = 4936
f19 = 4936
Figure 2 The 18th and 19th modes of the shell when t = 0.01
- As the thickness decreases, the number of longitudinal half waves remains at 1 (N = 1) for an increasing number of frequencies, and the number of circumferential waves increases. For the very thin shell (t = 0.00001), the lowest frequency corresponds to one longitudinal half wave and 14 circumferential full waves. For the moderately thick shell, the lowest frequency corresponds to N = 1 and M = 2 only, see the movie at top.
- For the moderately thick shell, we have one torsional mode, which does not exist among the lowest 20 frequencies/mode shapes of the thinner shells, see the movie in Figure 3.
Figure 3 The torsional mode for the shell with t = 0.01; originally
straight lines are used to highlight the torsional behavior.
- Of course, because of symmetry we have almost all frequencies appearing in pairs. However, these pairs of frequencies also become closer in spacing as t decreases. For example, whereas for t = 0.01 we have f1 = 1701 and f3 = 2184, we have for t = 0.00001 the values f1 = 74.78, f3 = 74.79. The top movie above shows the mode shapes corresponding to f1 = 1701 and f1 = 74.78.
These changes in physical behavior with a change in shell thickness are due to the fact that the bending energy scales cubically with t, whereas the membrane and shear energies scale linearly with t, while these energies are fully coupled through the displacement components [1, 2]. This coupling does not exist when considering other structures, e.g. simple straight beams or plates, and therefore the dynamics of these structures is quite different. For some analytical insight into the above observations, see Ref. .
These changes in physical behavior require in finite element analysis that the mesh used for solution of a dynamic response must depend significantly on the thickness of the shell. For example, whereas for the moderately thick shell (t = 0.01) the number of elements used circumferentially could be only twice the number used for the longitudinal direction, this ratio should be much larger for the very thin shell (or many elements could be used in both directions, which would be computationally more expensive).
In the above, we throughout considered linear analysis conditions. In nonlinear analysis, the same observations hold but additional effects can enter. In particular, in large deformation analysis, membrane stresses (in tension or compression) are usually present in the current configuration and can highly affect the dynamic behavior. We plan to demonstrate this effect later.
Since shell behavior can be complex and contain surprises, it is very important to use reliable and efficient shell elements and solution capabilities when analyzing shells in engineering practice — ADINA is very attractive in this regard.
- S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959.
D. Chapelle and K.J. Bathe, The Finite Element Analysis of Shells — Fundamentals, Springer, 2003.
- E. Artioli, L.B. da Veiga, H. Hakula and C. Lovadina, "On the asymptotic behaviour of shells of revolution in free vibration", Comput. Mech., 44:45-60, 2009.
Shells, dynamics, frequencies, mode shapes, shell thickness, sensitivity, membrane, bending, shear, coupling