Engineering Transactions,

**8**, 1, pp. 179-186, 1960### Wpływ Odkształcalności Przekroju Poprzecznego na Sile Krytyczna Wyboczenia Skrętnego Pręta Dwuteowego

The object of this paper it to determine the influence of deformability of the cross-section of an axially compressed double-tee bar on the length l on the critical force P for torsional buckling expressed by the Eq. (10). It is assumed that the bar has hinged supports, and that the end cross-sections are free to warp and are provided with rigid diaphragms. It is further assumed that the web is the only deformed element (of Fig. 2). The flanges are treated as bars, the equilibrium equations (1) being applicable. The deformation of the web as a plate is assumed in the form (3). Substituting the values of the deformations wo(x), Po(cx) and the end reactions m(ac) and r(x) according to (5), in (1), we obtain a system of two homogeneous equations (6), the determinant of which yields the buckling condition in the form of the transcendental equation (7), where the unknown is the dimensionless coefficient kkr = δer|E representing the ratio of the critical stress to YOUNG's modulus. If t/δ -> 0, the root of this equation in which we are interested tends to the value given by the Eq. (9), which corresponds to the flexural buckling of a flange.

Further, an example of iteration computation of the kpr coefficient is treated.

The iteration procedure consists in solving the quadratic equations (12), where, for the first approximation, we take the coefficient k, according to the Eq (11) for torsional buckling of a bar with undeformable cross-section. The convergence of the iteration is relatively good. In order to obtain numerical results sufficiently accurate for practical purposes, linear interpolation of the transcendental equation between Rm and k, is sufficient. Tables 1 and 2 represent the computation results. It is seen that the influence of deformability of the cross-section does not depend distinctly on the length of the bar, but increases rapidly for increasing parameter d|t. In rolled profiles, this difference should not exceed two per cent; in welded or composite cross-sections with very thick flanges it may be greater and exceed even 10 per cent.

Further, an example of iteration computation of the kpr coefficient is treated.

The iteration procedure consists in solving the quadratic equations (12), where, for the first approximation, we take the coefficient k, according to the Eq (11) for torsional buckling of a bar with undeformable cross-section. The convergence of the iteration is relatively good. In order to obtain numerical results sufficiently accurate for practical purposes, linear interpolation of the transcendental equation between Rm and k, is sufficient. Tables 1 and 2 represent the computation results. It is seen that the influence of deformability of the cross-section does not depend distinctly on the length of the bar, but increases rapidly for increasing parameter d|t. In rolled profiles, this difference should not exceed two per cent; in welded or composite cross-sections with very thick flanges it may be greater and exceed even 10 per cent.

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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

A. CHUDZIKIEWICZ, Wpływ odkształcalności przekroju poprzedzanego pręta cienkościennego na sile krytyczną Eulera, Rozpr. Inzyn. 1, 8 (1960).

. J. RUTECKI, Wytrzymałość konstrukcji cienkościennych, Warszawa 1957.