In modern technology use is made of thick-walled vessels of brittle materials such as concrete which are subjected to significant internal pressures. As a result of the poor resistance of brittle materials to tension, in the installation of such vessels those zones of the material in which tensile stresses from service loads are expected are subjected to preliminary strain (compression) using high-strength reinforcement. The amount of the preliminary compressive stresses, according to the data of [1], may be as high as 5 kgf/mm 2 in the tangential, 4 kgf/mm 2 in the axial, and 0.25 kgf/mm 2 In the radial directions. The figures given are an indication of the fact that the material of a structure under consideration may be in the limiting condition even under construction conditions if the ratios of the strength of the material, the dimensions of the structure, and the preliminary compressive stresses are not the optimum. To prevent the start of the limiting condition during the period of preliminary stress, it is necessary to establish the maximum calculated stresses which would not involve the danger of crack formation or failure. This problem has independent interest for designing transverse reinforced tubular design elements. Figure 1 shows the stressed condition of a vessel without heads loaded by a uniform external radial pressure and by axial compressive stresses from a centrally applied axial force. Under the action of these loads the vessel material will experience the complex stressed state. At the same time, on the internal surface of the vessel there is diaxial nonuniform compression, schematically shown in the center of Fig. 1. The remaining fibers of the walls undergo triaxial nonuniform (it may be called etlipsotdal) compression (plan in the left portion in Fig. 1). It may be assumed that the axial and circular (tangential) stresses are distributed in the plastic stage, that is, uniformly, since the limiting condition of the structure is considered and the radial stresses, as a result of their smallness, are in the elastic stage (according to the Lame relationship). In all cases the compressing stresses are taken as positive. To establish the values of the limiting stresses methods of determining the strength of the material in diaxial nonuniform and ellipsoidal compression are necessary. Let the strength of the material in diaxial uniform compression follow a piecewise linear relationship of the form (,~ - og) (o~ - oD (I ,~,- ,,~l - %) = o, 0~<%, %<o~, I trl--ad =%;