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Cylinders and spheres are often used to contain hot or cold fluids. A temperature gradient across the vessel wall results in thermal stresses. This paper presents equations for means and standard deviations of thermal stresses developed in cylinders and spheres. Inherent in these equations are the facts that: (a) design variables are generally characterized by spectra of values, and (b) a small, but finite probability of failure must be recognized in any design. By coupling the stresses due to a thermal gradient as calculated by the equations in the paper with the strength available in a material, reliability (or the alternative probability of failure) can be calculated. Conversely, for a specified reliability the appropriate size or heat flux can be determined. Appropriate illustrations of application of these equations are provided by tables and figures. The difficulty of relying on a factor of safety is demonstrated.

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