Engineering Transactions, 8, 3, pp. 423-461, 1960

### Ogólna Teoria Stateczności Prętów Cienkościennych z Uwzględnieniem Odkształcalności Przekroju Poprzecznego. Cześć I. Pręty O Prostym Przekroju Poprzecznym

A. Chudzikiewicz
Politechnika Gdańska
Poland

This is a continuation of the former paper by the present author, Refs.[2]-[4]. The subject matter is a theory of thin-walled bars constituting a generalization of the known theories of primary and local buckling. These theories 'are not accurate because they do not take into consideration the influence of deformability of the cross-section of the deformation of the bar and, on the other hand, the influence of the deformations of the bar on the deformations of its walls. The assumptions are as follows: the elastic range, axial compression, the polygonal cross-section and constant thickness of every wall. It is assumed further that the ends of the bar are stiffened by thin diaphragms preventing the deformation of the end cross-sections and that the supports are hinged ones and enable free warping of the end cross-sections. The polygonal cross-sections are classified into two groups. Characteristic* of the first sections, called simple (Fig. 1) is that their deformation causes directly only the displacements normal to the wall (the tangential components being equal to zero). In the cross-sections of the second group (Fig. 2) called composite displacements tangential to the wall are also possible. Cross-sections of the first group are discussed in detail in the present paper. The second group will constitute the subject matter of a separate paper. For each deformed wall y of the bar (Fig. 6,7), treated separately, the equilibrium equation (2.3) is established, P, denoting part of the compressive force acting on the wall and p,--the distributed reactions of the adjacent walls. The shearing force T, is expressed as use of the fact that for a a sum of the shear stresses t making simple cross-section they do not depend directly on the deformation of the cross-section and are given by the Eq. (2.1) for indeform- able cross-sections. Thus, finally, we obtain equations of the type (2.7) in which the displacement components E, n appear as well as the rotation q of the cross-section and the displacements of the walls t, which may always be expressed in terms of the quantities E, n, q. In addition, the equations of equilibium (2.10) are established for the transversal bending moments acting along the edges.
The reactions Prr Mys between the walls cons tituting the reactions of a plate compressed in one direction with longitudinal edges undergoing edge displacements 4, 0 (Fig. 11,12), can always be expressed in terms of 5, N, P and the rotation angles Y of the edges. The relevant formulae (3.8), (3.9), (3.10) are obtained by solving the Eq. (2.9). As a result there remain always in all the equations of the type (2.7) and (2.10) the three unknown displacement components of the bar 5, n, q and the rotation angles yn, determining the de- formation of the cross-section. Setting the determinant of the system of homogeneous equations thus obtained equal to zero, we obtain the buckling condition in the form of a transcendental equation, in which the unknown is the dimensionless coefficient k = G|E (where o is the critical stress).
In Sec. 4 detailed analysis of stability is given for the most important types of cross-section. In Sec. 5 plate buckling of the walls is considered as a particular case, which may be obtained by assuming § = n = Q = 0 in the general equations obtained. In Sec. 6 numerical results are given and a discussion for flexural buckling of a channel section. These results are compared with those of the Ref. [4] where indeformable flanges were assumed.
The conclusions may be formulated thus: 1. The influence of the deformability of the cross-section on the critical force in rolled profile bars is insignificant and amounts to a fraction of one percent. 2. In bars with very flexible web and broad flanges this influence may be greater, but in practice it exceeds a few per cent only in rare cases. 3. The influence of the deformability of the cross-section increases rapidly with decreasing the length of the bar. In very short bars it may reach theoretically a few tens per cent. In these cases the critical force does not differ practically from the critical force for plate buckling of the walls computed for n = 1 (Fig. 28, Table 11); 4. It may practically be always assumed that the flanges are indeformable. The errors are significant only in the case of very thin flanges. 5. The influence of bar deformations on local buckling is insignificant and may be disregarded.

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#### References

[in Russian]

A. CHUDZIKIEWICZ, Wpływ odkształcalności przekroju poprzecznego pręta cienkościennego na siłę krytyczną Eulera, Rozpr. inyn., 1, 8 (1960).

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A. CHUDZIKIEWICZ, Giętne i gietno-skrętne wyboczenie pręta ceowego o odkształcalnym przekroju poprzecznym, Rozpr. inzyn., 2, 8 (1960).

F. BLEICH, Buckling Strength of Metal Structures, New York 1952.

E. STOWELL, C. LIBOVE, E. LUNDQUIST, G. HEIMERL, Buckling Stresses for Flat Plates and Sections, Proceedings ASCE, 77, 1951.

R. DABROWSKI, Praktycznie ważne przypadki wyboczenia skrętnego prętów cienkościennych, Arch. Inzyn. ladow. 1, 1956.