Engineering Transactions, 10, 3, pp. 567-581, 1962

Stan Graniczny Pręta Jednocześnie Skręcanego i Rozciąganego przy Dowolnym Kształcie Przekroju

M. Wnuk
Politechnika Krakowska

The problem of combined stress in the purely plastic state resulting from a simultaneous action of a longitudinal force N and a torque Ms is solved in displacemonts, the Hill stress equation being not used. Assuming the existence of a perfect elastic-plastic incompressible material and making use of the Hencky-Ilyushin strain theory, the basic equation (2.11) is derived in displacements. This is an elliptic non-lincar equation with a non-homogeneous boundary condition (2.13). In Sec. 2 general equations are given for the stresses Tac, Tay, 0z and the corresponding equations for the outer forces N. M which are certain functional expressions of the dimensionless warping of the cross-section Y = p (x, y). Equation (3.5) containing the unknown function 3p represent the accurate paramstric equation of the limit curve in the plane Ms N for any profile. The possibility of approximate solution for the limit curve is obtained by assuming y = 0.

The basic equation in displacements is soived by means of the perturbation method taking into consideration the influence of torsion on tension and showing that it is possible to make use of all the solutions of the Laplace equation 7/2 Wo = 0, known from the theory of elasticity. Con- fining ourselves to the first approximation, simple effective equations (4.22), (4.23) are obtained rulating the coefficient a characteristic of the form of the limit curve (4.20) for Ms & n with the geo-
metric characteristics of the profile under consideration. Finally, applying Hermitian interpolation, the applicability range of the Eq. (4.20) is extended to the entire 0 < ms < 1, 0 < n < 1 region (where ms is the dimensionless torque and n- the dimensionless longitudinal force) thus obtaining the full equation of the limit curve f (ms, n) = 0, (5.2), valid for any profile, The examples given in conclusion concern the limit load (ms, n) of a bar in the form of any regular polygonal cross-section: the approximate formula (5.6) proposed gives in the particular case where the number of sides of the polygon p -> o the known accurate equation of the limit curve for the circular bar.

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


[in Russian]

F.A. GAYDON, On the Combined Torsion and Tension of a Partly Plastic Circular Cylinder, Quart. J. Mech. and Appl. Math., 1, 5 (1952).

A. E. GREEN, E. W. WILKES, A Note on Final Extension and Torsion of a Circular Cylinder of Compressible Elastic-Izotropic Material, Quart. J. Mech. and Appl. Math., 2, 6 (1953).

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

A. KOVAR, Theorie krouceni, Praha 1954.

[in Russian]

A. NADAL, Plasticity, New York-London 1931.

S. PIECHNIK, The Influence of Bending on the Timit State of a Circular Bar Objected to Torsion, Arch. Mech. Stos., 1, 13 (1961), 77-106.

S. PIBCANIK, M. ZYCZKOWSKI, On the Plastic Interaction Curve for Bending and Torsion of a Circular Bar, Arch, Mech, Stos., 5, 13 (1961).

[in Russian]

[in Russian]

M.A. SADOVSKY, A Principle of Maximun Plastic Resistance, J. Appl. Mech., June 1943.

B. SAINT-VENANT, Mémoir sur le torsion des prismes, Gosizdat. Fiz.-Mat. Literat., Moskwa 1961.

B.R. SETH, Final Elastic-Plastic Torsion, J. Math. Phys., 1, 31 (1952).

C. WEBER, W. GÜNTHER, Torsionstheorie, Braunschweig 1958.

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