8, 1, pp. 47-61, 1960
We consider a thin-walled bar with closed quadratic cross-section (Fig. 1). The possibility is investigated of stability loss where all the walls buckle in their planes, the axis of the beam remaining rectilinear. The cross-section of the bar is thus deformed. It is assumed that the end sections x = 0, x = l are stiffened by ribs and are free to warp. The starting point is the Eq. (2.6) for a wall with the moment of inertia I compressed by the force PJ4 and loaded by the two reactions r. These reactions are due to the neighbouring walls constituting plates subject to deformation (Fig. 2), and are calculated from the Eq. (2.1) for a plate compressed in one direction. The critical force is obtained from the transcendental equation (2.10). This force may be expressed in terms of the critical force of Eulerian buckling (2.11) by using the coefficient x, which is computed from the transcendental equation (2.12). The dimensions of the bar, for which we have x = 1, are represented in Table 1. For % = 0,5 there is no interaction between the walls which suffer independent buckling. In Sec. 3, an approximate equation (3.5) is obtained by means of the energy method, assuming the deflection surface of the wall in the form (3.2). The values of the coefficient x obtained from that equation are shown in Table 3. The errors do not exceed one percent. In Sec. 4, the transcendental equation (4.4) is derived for the plastic range on the basis of shear modulus the hypothesis and BLEICH's theory. The results obtained lead to the conclusion that the critical load may, according to the theory presented, be considerably lower than the EULERIAN force, but that for the usual dimensions of steel bars this case in unlikely to occur either in the elastic or in the plastic range. On the other hand, for bars made of materials with low modulus of elasticity and high elastic limit the buckling form under consideration may prove to be of some importance also in practical cases.
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).
S. TIMOSHENKO, Theory of Elastic Stability, New York 1936.
F. BLRICH, Buckling Strenght of Metal Structures, New York 1952.
J. RUTECKI, Wytrzymałość konstrukcji cienkościennych, Warszawa 1957.