Engineering Transactions, 16, 2, pp. 233-260, 1968

### Obliczanie nośności granicznej skręcanych prętów o dowolnej niejednorodności poprzecznej

M. Galos
Politechnika Krakowska
Poland

The problem under consideration is that of free plastic torsion of a prismatic bar characterized by transversal plastic nonhomogeneity of general type. It is assumed that the nonhomogeneity function k = k (x, y) (the yield limit for pure shear) and its derivatives δk/δx, δk/δy, are continuous. in component subregions of the cross-section of the bar. The discontinuities admissible at the border are those of the nonhomogeneity function itself and its derivatives. The problem is solved by introducing the stress function p. The differential equation (3.9) of the line of greatest slope of the stress function D is obtained. Strictly speaking the term of line of greatest slope is used to denote not the line itself but its projection on the cross-section of the bar. As an equation of the characteristic lines the Eq. (3.9) was derived by J. A. KUKO, [6]. The differential equation of the line of the greatest slope obtained by another method and discussed in the present paper was communicated by the present author at the International Conference of the IPPT, PAN at Kolobrzeg in 1966, therefore before Ref. [6] was published. Effective methods are proposed for the determination of discontinuity lines of stress. The paper contains also equations for the determination of the limit moment. The effective equation for the limit load is derived making use of the generalized Nádai analogy, [9], which states proportionality between the limit moment and the volume below the surface Φ (x, y). By appropriate selection of curvilinear coordinates the computation of the volume below the surface Φ is reduced to a very conveniant integration procedure of a function of two variables. The methods used can be also made use of for solving associate problems, such as the problem of plastic torsion of circular bars of variable diameter, made of material of circularly symmetric nonhomogeneity and other problems. The work is illustrated by a numerical example concerning a bar, the cross-section of which is shown in Fig. 11. The material is assumed to be of plastic nonhomogencity of the type k = k_0 e^(a^((x^2+ y^2)) ). The results of Secs. 2 to 11 are used to determine the lines of discontinuity of stress, the lines of greatest slope of the function Φ, the function Φ itself, the limit torque M and the deformation of the cross-section of the bar. The value of the limit torque is compared with the analogous value for a bar made of a homogeneous material.
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#### References

W. FREIBERGER, W. PRAGER, Plastic twisting of thick-walled circular ring sectors, J. Appl. Mech., 23 (1956), 461-463.

M. GALOS, M. ŻYCZKOWSKI, Analityczna metoda obliczania nośności granicznej prętów skręcanych, Rozpr. Inżyn., 2, 12 (1964), 267-296. ang.: Bull. Acad. Polon. Sci., Série Sci. Techn.,

(1964), 69-78.

M. GALOS, On analytical calculation of the limit carrying capacity of isotropic and non-homogeneous bars under torsion, Bull, Acad. Polon., Sci., Série Sci. Techn. 5, 12 (1964), 241-247.

R. HILL, On the problem of uniqueness in the theory of a rigid-plastic solid, J. Mech. Phys. Solids, 5 (1956), 1-8.

[in Russian]

[in Russian]

A.I. KUZNIECOW, The problem of torsion and plane strain of non-homogeneous plastic bodies, Arch. Mech. Stos., 4, 10 (1958), 447-462.

J. MANDEL, Sur les deformations de la torsion plastique, C. R. Acad. Sci., Paris 222 (1946), 1205-1207.

A. NÁDAI, Der Beginn des Fliessvorganges in cinem tordierten Stab, ZAMM, 6, 3 (1923), 442-454.

W. PRAGER, P.G. HODGE, Theory of Perfectly Plastic Solids, Wiley, New York 1951. Tłum. ros.: IL, Moskwa 1956.

J. RYCHLEWSKI, Plastic torsion of non-homogeneous bars analyzed in curvilinear coordinates, Bull. Acad. Polon. Sci., Série Sci. Techn., 7, 12 (1964), 469-474.

J. RYCHLEWSKI, Plastic torsion of bars with jump-non-homogeneity, Acta Mechanica, 1, 1 (1956), 35-53.

M. A. SADOWSKY, An extension of the sand heap analogy in plastic torsion applicable to cross-section having one or more holes, J. Appl. Mech., 2, 8 (1941), 166-168.

[in Russian]