Engineering Transactions, 8, 1, pp. 103-119, 1960

### Wpływ Odkształcalności Przekroju Poprzecznego Pręta Cienkościennego na Siłę Krytyczną Eulera

A. Chudzikiewicz
Politechnika Gdańska
Poland

The object of this paper is to determine the influence of the deformation of the cross-section (Fig. 1) of a compressed double-tee bar of length on the EULERIAN critical force Pe in the elastic range. Hinged end supports and rigid end diaphragms preventing the deformation of the end cross-sections are assumed. It is also assumed that the critical stress obtained will be less than the critical stress of local buckling. Since we have always 0 >t, it is assumed that the horizontal elements are not deformed in the plane of the cross-section. Therefore, the vertical part is the only part to undergo deformation, its deflection being w(×,y) (Fig. 3). In further considerations, the horizontal elements are treated as bars subjected to bending, torsion and compression, and the vertical element as a plate subjected to bending and compression. Two equilibrium equations must be satisfied: for bending, (2.1), and for torsion, (2.2). For the vertical part the familiar Eq. (2.4) must be satisfied.
The solution is assumed in the form (2.6). It satisfies the boundary conditions (2.3) for æ = 0 and x = l, and the conditions (2.5) at the edges x = 0 and x = l. There remain the compatibility equations of strain and displacement to be satisfied at the edges y = + h/2. be (2.10) in (2.1) and (2.2). Hence the buckling condition is obtained in the form This may done by substituting of the transcendental equation (2.14). In Sec. 3, an iteration method for calculating the critical force is described. The EULERIAN stresses are assumed as the first approximation or, in other words, ky = kp according to (3.1). The quantities thus obtained are then assumed to constitute the load of the horizontal elements. Thus, the second approximation kg is obtained as the lower of the roots of the quadratic equation (3.3). Further approximations are calculated from the quadratic equation
(3.4). The iteration is not rapidly convergent, but a linear interpolation according to Fig. 5 yields a good result.
In Sec. 4, are presented the calculation results.
Table 1 represents the dependence of the diminishing of the EULERIAN force on the length / of the bar, Table 2 showing the dependence on the para- meter Ut, and Table 3-on the parameter d/t. From a discussion, it follows that for rolled girders this influence is expressed by fractions of one per cent (Table 3); for welded girders it may exceed one per cent and with decreasing length of the bar it increases rapidly (Table 1); but then, the critical stress exceeds the elastic limit and the stress of local buckling.

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#### References

H. WAGNER, Verdrehung und Knickung von offenen Profilen, Festschrift 25 Jahre T.H. Danzig, Gdañsk 1929.

A. OSTENFELD, Politecknisk Laereanstalts Laboratorium for Bygningsstatik, Meddelelse nr 5, Kopenhaga 1931.

R. KAPPUS, Drillknicken zentrisch gedrückter Stäbe mit offenen Profil im elastischen Bereich, Luftfahrt-Forschung, 1937.

F. BLEICH and H. BLEICH, Bending Torsion and Buckling of Bars Composed of Thin-Walls, Preliminary Pub. 2 Cong. Intern. Assoc, Bridge and Structural Engen, Berlin 1936.

[in Russian]

S. TIMOSHENKO, Theory of Elastic Stability, New York 1936.

A. CHUDKIEWICZ, Utrata stateczności przez zniekształcenie przekroju poprzecznego pręta cienkościennego, Rozpr. inzyn., 1, 8 (1960).

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

R. DABROWSKI, Praktycznie ważne przypadki wyboczenia skrętnego prętów cienko-ściennych, Arch. Inzyn. ladow., 1 (1956).

J. NOWIŃSKI, Theory of Thin-Walled Bars, Appl. Mech. Rev. 4, 12 (1959).

J. RUTECKI, Niestateczność pręta cienkościennego z uwzględnieniem odkształcenia profilu, Arch. Mech. stos., 3/4, 3 (1951).

J. RUTECKI, Wytrzymałość konstrukcji cienkościennych, Warszawa 1957.