The object of this paper is to investigate the influence of the deformability of the cross-section of a channel on the critical force for flexural and flexural-torsional buckling. It is assumed that the material is elastic, the end supports hinged and enabling free warping of the end cross-sections which are stiffened by means of trans-versal diaphragms preventing the deformation of the cross-section. It is further assumed that the flanges are indeformable in the plane of the cross-section. The displacements of points of a cross-section are determined by four quantities (Fig. 4): two displacement components 5, n of the cross-section, its rotation g and its deformation characterized by the angle y, and by the real compressive stress o. Each wall of the bar is considered separately (Fig. 3). The displacements are expressed by the Eqs. (2.1). The Eqs. (2.4) and (2.8) represent the shearing forces (Fig. 6) acting in the flanges and the web of the channel cross-section due to the deformation of the bar as a whole. These forces depend only on the quantities ξ, n, ϕ, because the angle y does not influence directly the distribution of the stresses o and t. They enter, together with the reactions Y., T., m, (Fig. 3), the equations of equilibrium of the flange (2.5) and the web (2.9). In Sec. 3, the reactions 1, m are computed for a plate compressed by the stresses o = kE depending on the edge displacements 4, 0 (Fig. 7). In Sec. 4, the flexural buckling is considered. In view of the symmetry,; it is assumed that N= p = 0 and that the deformation of the web is symmetric (Fig. 7a). On substituting the reaction (4.4) in the Eqs. (4.1), and assuming the solution in the form (4.7), we obtain the transcendental equation (4.8), in which the dimensionless coefficient key Or/E is unknown. The results are collated in Table 1. The decrease of the critical force in relation to the Eulerian force is from 0.69% to 78.9% and increases violently with increasing width of the amount web. Next, an iteration procedure is considered according to the Eq3. (4.14) and (4.15). The successive approximations of the coefficient k are collated in Table 2. In Sec. 5, combined flexural-torsional buckling is considered. We assume E = 0 and antisymmetric deformation of the web (Fig. 7b). The transcendental equation is obtained from (5.5). For k/b = 1,25 the coefficient kar= 1,90 · 10-3 1S computed while from the familiar equation (5.6) we obtain k = 2,02 · 10-3, therefore the decrease of the critical force amounts to 5,94% and is considerably smaller than for the Eulerian buckling. The numerical results lead to the conclusion that in typical cases the influence under consideration is small, but in cross-sections having flanges very rigid in comparison to the «vertical» element, it may be of practical meaning.
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).
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