©2012 Civil-Comp Ltd
R. Okhovat and A. Boström
Department of Applied Mechanics, Chalmers University of Technology, Göteborg, Sweden
Keywords: spherical shell, shell equations, dynamic, eigenfrequency, power series.
full paper (pdf) -
Shells are important structures in many branches of engineering and have therefore a number of different types have been investigated. Spherical shells appear in some applications, and some dynamic shell theories have thus been developed for this case. Most of these theories depend on more or less ad hoc kinematical assumptions and, or other approximations. Here the dynamic equations for a spherical shell are derived by using a method developed during the last decade for bars, plates, and beams. It has, in particular, been developed for a number of different plate structures, such as anisotropic, layered, and piezo-electric plates. The main advantage with the method is that it is very systematic and can be developed to any order. It also seems that the resulting structural equations are asymptotically correct to any order.
To be specific, the dynamic equations for a spherical shell made of a homogeneous, isotropic material are derived. The shell is assumed to be thin relative to the radius of the sphere and relative to relevant wavelengths. The starting point is a power series expansion in the thickness coordinate (around the mid-surface of the shell) of the displacement components. Here the expansion coefficients are functions of the angular coordinates and time. These expansions are inserted into the three-dimensional elastodynamic equations. Each power in the thickness coordinate then results in a recursion relation among the expansion functions. These recursion relations can be used to express all higher-order expansion functions in terms of the six lowest-order ones. It is noted that the thinness of the shell is not exploited during this process and that there are no approximations involved so far. The power series expansions of the displacement components are inserted into the stress-free boundary conditions on the two spherical surfaces of the shell and these then become six power series in the shell thickness. Eliminating all but the six lowest-order expansion functions with the help of the recursion relations finally gives six dynamic equations for the shell. These can, in principle, be truncated to any order in the shell thickness, but in practice only very low orders are of interest.
To investigate the properties of the resulting shell equations the eigenfrequencies for the spherical shell are computed for two simple cases. Comparisons are made with exact three-dimensional calculations and membrane theory. The first case is the purely radial vibrations where the agreement between all theories is excellent for the first mode, but less so for the second mode (where membrane theory does not apply). But it should be remembered that the second mode has a radial dependence that it is hard to reproduce with a simple power series with a few terms and that the shell is no longer thin (in terms of wavelengths) for the second eigenfrequency. The second case that is investigated is the torsional modes and exactly the same comments apply for this case.