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

### Stan Graniczny Pręta Jednocześnie Skręcanego i Rozciąganego przy Dowolnym Kształcie Przekroju

M. Wnuk
Politechnika Krakowska
Poland

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.

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