Engineering Transactions, 5, 1, pp. 51-96, 1957

### Nośność Graniczna Niejednorodnych Wirujących Tarcz Kołowych

M. Życzkowski
Instytut Podstawowych Problemów Techniki PAN
Poland

The problem is solved using the two yield conditions most commonly used - energy-of-distortion condition, and maximum shear stress. It is shown in Art. 2 that, subject to some concerning the form of the very weak assumptions of limit state of rotating yield condition, the solution of discs of varying thickness the problem to the problem of a disc of constant thickness can always be reduced with modified non-homogeneity, characterized, in the circularly symmetrical case, by the dependence of the yield point @ and the specific weight y on the radius r. In the case of a plastically non-homogeneous disc, the solution is particularly simple if the theory of maximum shear stress is used. This is because we have 050,508 throughout the entire dise, and the yield condition (3.1.1) is linear. In Art. 3 necessary and sufficient condicase of plastically non-homogeneous discs. tions are established enabling the condition (3.1.1) to be used also in the simple equation (3.13), which is obtained by determining the limit angular velocity of the disc, is of limited applicability, as being based on the yield condition mentioned above [the necessary conditions (3.29) and (3.37) should be satisfied]. An example is given in which use is made of several simultaneous equations of the type (3.1), determining the yield condition based on the maximum shear stress theory. The problem of limit state of a rotating disc using the energy-of- distortion theory is stated in Art. 4 and solved in Art. 5. The (approximate) method of «assumption of accurate solution» constituting a generalization of the author's method of «assumption of accurate equation», [26], is employed, and example is given of the application of this method when the distribution of the yield point is determined by the Eq. (5.15).
In Art. 6, the stresses are expressed in trigonometric form in the case of plastically non-homogeneous bodies. In Art. 7, this method is used for the purpose of solving the problem under consideration. It is concluded that representation of the stresses in trigonometric form does not here present any essential advantage. In Art. 8 special consideration is given to the problem of a plastically non-homogeneous disc, of which the limit state has been calculated by V. V. Sokolovsky by means of the finite difference method. It is shown that the value obtained by that author is somewhat too low, and
that the figure 1.79 should be replaced by 1.805. The above considerations are confined to the determination of the
limit angular velocity and the stress distribution. They retain their validity both in the Hencky-Iliushin theory of small elastic-plastic deformations, and in the Prandtl-Reuss and Levy-Mises theory of plastic flow. Disc deformation is considered in Art. 9, the Hencky-Iliushin theory being used. First, general equations for strain in the circularly symmetrical case are derived for a known stress distribution. These equations (9.4), which are valid for a non-homo-
geneous compressible medium, are considerably simplified if incompressibility is assumed. They are the Eqs. (9.5) expressed in the dimension- less form or, if the trigonometric form is used, the Eqs. (9.6). From these equations, it follows that the function 20g- 0, must be non-negative over the entire interval 0 ≤ R. This involves some limitation of the range of applicability of the previous considerations. If this condition is not satisfied, it means that the disc will suffer destruction before it passes entirely into the plastic state. The limit state of full discs is considered in this paper. The difference for discs with axial borings free from load is not, however, serious.

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