Engineering Transactions, 4, 3, pp. 353-366, 1956

Wyboczenie Pręta Pryzmatycznego jako Zagadnienie Dynamicznej Teorii Plastyczności

J. Madejski

The buckling of a prismatic bar in the plastic region (strictly speaking its breakdown under axial forces) was investigated experimentally among others by M. Broszko who presented his results in Ref. [2]:  These results led Broszko to the conclusion that if the slenderness of the bar is below its critical value the destruction of identical test pieces is more rapid for greater loads. The problem is therefore influenced by time, in other words, the period during which the load acts. This is in agreement with S. Timoshenko's opinion, [3], that the plastic buckling is a phenomenon of unstable nature. The minimum value of buckling load below which the strength of the test piece (with slenderness below its critical value) is unlimited in time is called by M. Broszko the effective buckling stress. From the foundations of the dynamical theory of plasticity, treated by the author in Ref. [4], it follows that the effective buckling stress coincides with the true yield point (i.e. the stress above which the temperature of the test piece starts increasing). If the test piece has been given a sufficient treatment the effective buckling stress (the true yield point) is equivalent to an illimited fatigue strength.
If the stress exceeds the true yield point, the test piece will be broken after a finite period. The foundations of the dynamical theory of plasticity, [4], have been used to establish the relation between the stress for which the first fibres break and the slenderness, the period during which the loads acts being constant. The following simplifications are introduced in these considerations: (1) the test piece has a particular curved form before the load is applied; (2) the time variability of the load corresponds to the Heaviside function; (3) the phenomenon is isothermal and tautothermal; (4) the problem is linearized due to a simplified formula for the radius of curvature. The considerations are confined to the case in which the stresses in the whole cross-section and during the whole period of action of the load are higher than the true yield point. The elastic-plastic problem can also be solved using analogous methods.

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