Engineering Transactions, 12, 2, pp. 267-296, 1964

Analityczna Metoda Obliczania Nośności Granicznej Prętów Skręcanych

A. Gałos
Politechnika Krakowska

M. Życzkowski
Politechnika Krakowska

Owing to the analogies of Nadai and Sadowsky, the problem of calculation of the limit load of an isotropic homogeneous bar subject to torsion is reduced to that of computing the volume of a solid of uniform slope (a «sand hill»). The aim of the present paper is to derive general equations determining this volume. In Sec. 2 are given equations of discontinuity lines of stresses corresponding to regular arcs of the contour, one concave singularity and two concave singularities. These are, respectively, Eqs. (2.3), (2.4), (2.17) and (2.18). The volume of the sand-hill or part of it is determined by the respective general Eqs. (3.5) or (3.6), (3.9) or (3.13) and (3.15). As an example are given solutions for elliptic profiles, a semicircular profile, circle with a circular incision or with two circular incisions, a profile bounded by two cycloidal arcs and a cardioid.
The results are represented in the form of tables and graphs. In Sec. 6, the equations derived are applied to the computation of the limit load of anisotropic bars, because it has been shown by W.O. GEOGDŻAJEW, [S], that this problem can be reduced by substituting (6.4) to that of an isotropic bar with transformed contour. In Sec. 7, the equations derived are applied to the computation of the limit load of a non-homogeneous bar subject to torsion if the yield point depends only on the distance of the point from the contour of the section; this type of non-homogeneity may have a practical value in the case of surface heat treatment. The case of discontinuous non-homogeneity, is also considered.

Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


R. ACKERMANN, Böschungsstrahlen und Böschungsflächen, Dissertation, Halle 1913.

[in Russian]

F.A. GAYDON, On the combined torsion and tension of a partly plastic circular cylinder, Quart. J. Mech. Appl. Math., 1, 5, (1952), 29-41.

F.A. GAYDON, H. NUTTALL, On the combined bending and twisting of beams of various sections, J. Mech. Phys. Solids, 1, 6 (1957), 17-26.

[in Russian]

De la GOURNERIE, Traité de geometrie descriptive, V. II, Paris 1885, 54.

[in Russian]

R. HILL, The Mathematical Theory of Plasticity, Clarendon Press, Oxford 1950.

R. HILL, M.P.L. SIEBEL, On combined bending and twisting of thin tubes in the plastic range, Phil. Mag., 7, 42 (1951), 722-733.

R. HILL, M. P. L. SIEBEL, On the plastic distortion of solid bars by combined bending and twisting, J. Mech. Phys. Solids, 1, 1953, 207-214.

E. O. IMEGWU, Plastic flexure and torsion, J. Mech, Phys. Solids, 2, 8 (1960), 141-146.

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

[in Russian]

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

A. NÁDAI, Theory of Flow and Fracture of Solids, McGraw-Hill, New York 1950. Tłum. ros.: IL, Moskwa 1954.

S. PIECHNIK, The influence of bending on the limit state of a circular bar subject to torsion, Arch. Mech. Stos., 1, 13 (1961), 77-106.

S. PIECHNIK, M. ZYCZKOWSKI, On the plastic interaction curve for bending and torsion of a circular bar, Arch. Mech. Stos., 5, 13 (1961), 669-692.

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

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

F. SCHILLING, Über die Böschungsflächen mit Kegelschnitten als Basiskurven, ZAMM, 3, 3 (1923), 197-217.

[in Russian]

Z. SOBOTKA, Theorie Plasticity, t.I., Nakl. CSAV, Praha 1954.

[in Russian]

M. WNUK, Stan graniczny pręta jednocześnie skręcanego i rozciąganego przy dowolnym kształcie przekroju, Rozpr. Inzyn., 3, 10 (1962), 565-581.

M. WNUK, Oszacowanie krzywej nośności granicznej przy jednoczesnym skręcaniu z rozciąganiem, Rozpr. Inzyn., 1, 11 (1963), 179-200..

M. WNUK, Krzywe nośności granicznej dla jednocześnie skręcanych i rozciąganych prętów o róznych kształtach przekroju, Rozpr. Inzyn., 4, 11 (1963).

M. ŻYCZKOWSKI, Przypadek jednoczesnego rozciągania i skręcania pręta o przekroju kołowym w zakresie sprężysto-plastycznym, Rozpr. Inzyn., 2, 3 (1955), 285 -322.

M. ŻYCZKOWSKI, Powierzchnie graniczne w teorii wytężenia, Rozpr. Inzyn., 4, 9 (1961), 609-637.