The paper is concerned with simply supported shallow conical shells loaded through a central boss. The shell material is rigid-plastic and the relevant stress resultants are subject to a nonlinear yield condition. The critical load intensity at which the shell will begin to deform and the associated fields of stress resultants and plastic flow are to be determined. For this purpose, a problem concerned exclusively with stress resultants is formulated by using the equation of stress compatibility. The resulting nonlinear problem is solved, with the help of a digital computer, in cases where the yield-point state of stress is controlled, over the pertinent intervals covering the entire shell, by three distinct yield conditions, each corresponding to a given face of the appropriate four-dimensional nonlinear yield surface.

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