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The purpose of this report is threefold:(a) to derive a stress analysis of the sphere-cylinder structure that treats the two components with a minimum of overlap at their junction; (b) to present a computational procedure that employs only three geometrical parameters, so as to facilitate the calculation of critical stresses over a wide range of sizes and proportions, in other words, a large-scale parametric study, and (c) to include one or more tables of numerical values that can be used for the simplified calculation of stresses at any point in the neighborhood of the sphere-cylinder junction. All results are based on an internal pressure of 1 psi and a mean sphere diameter of 1 inch; values for other pressures and sizes are obviously in direct proportion. This study was undertaken as part of a general investigation of the reinforcement of openings in pressure vessels by the Pressure Vessel Research Committee of the Welding Research Council. The intent was to obtain stress values for radial cylindrical outlets in spherical pressure vessels that could be compared with experimental results in the same field. If, as a result of this comparison, the theoretical data appear to be reasonably valid, they can then be used as a basis for the estimation of stress conditions in openings of this type, over the whole range of sizes and proportions currently employed in industry.The commonly-accepted or so-called "classical theory" of axisymmetrical shells with constant pressure internal loading is the basis of the analysis which follows. In the solution of the cylinder equations, nothing more complicated than trigonometric and exponential functions is involved. Following Meissner's analysis of the spherical shell, it develops that the exact solution involves a hypergeometric series. In the writer's opinion, this solution is of little more than academic interest, since its convergence, although absolute, is extremely slow except for thick-walled vessels and small values of the colatitude . Esslinger's approximation (cot replaced by 1/) was used, for the results tabulated herein; this contains Bessel-Kelvin functions which are found in easily accessible tables with a high degree of accuracy. An estimate of the validity of this approximation at large values of is given in a later paragraph. It is recommended that, where more accuracy is desired in the range 50 < < 90, an asymptotic series solution be used; this degenerates in to the more familiar Geckeler approximation as approaches 90. Since the experimental program made use of the photoelastic method of analysis, with materials having the high value of Poisson's ratio commonly associated with plastics and elastomers, a value of 0.5 was adopted for all computed results throughout this report. It should also be noted that, since the final results have been reduced to dependence on not more than three parameters, no account is taken of the effect of outside fillets or inside corner roundings. The fillet will have a stress-raising effect on the outside surface which should be predictable by a simple semi-empirical formula based on curved-beam theory. At the same time, another correction is applicable due to the thickness changing effect of both the fillet and the inside corner; this can be either additive or subtractive. In the case of stresses in the outlet near its base, it may be expected to give an appreciable reduction.
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