Thermally Induced Vibrations of Concentric Thin Spherical Shells

Abstract

If a thin, unrestrained spherical shell is rapidly heated, large inertial hoop stresses may be developed which result in free oscillation. It has been shown that the dynamic stress amplitude is dependent upon the ratio of heating time to the natural period of oscillation as well as upon the maximum temperature. Since a free shell is rarely encountered in practice, the purpose of this study is to determine the dynamic response of a set of concentric spherical shells when the inner shell only is subjected to rapid, uniform, internal heat generation. The maximum number of shells chosen for analysis is three; however, the method is general and may be applied to systems containing as many concentric shells as desired. The results are presented in parametric form for the stresses in each shell and their dependence upon the material properties. In most reactor-design problems it is desired to maintain the integrity of any system; hence, it is assumed that the inner, heated shell always remains elastic. This represents the extreme stress condition, and may cause yielding of the outer shells. For constraint of the inner shell, the dynamic stresses are obtained for elastic motion and when the outer shell is allowed to flow plastically at constant stress. The special case of instantaneous heating and the effect of composite material properties upon stress amplitudes is considered in detail to provide useful design formulae.