Wpływ Przepon Pośrednich na Stateczność Pręta Cienkościennego

This is a continuation of the author's former papers, [1] and [2]. The object is to investigate the influence of an intermediate diaphragm on the stability of an axially compressed thin-walled channel. It is assumed that the bar is perfectly elastic, simply supported and its end cross-sections are free to warp. The diaphragms are assumed to be perfectly flexible, the walls are of uniform thickness and the flanges are indeformable in the plane of the cross-section.
The buckling described by the functions ξ(x) and ϕ(x) (Fig. 3) is considered. In view of the symmetry the equations are established for one flange only. In these equations A = bδ is the cross sectional area, I = δb3/12 - the moment of inertia of the flange, E-Young's modulus, S(1) the static moment of the flange in relation to the x axis, p and m are the full loads of the flange after the stability loss (Fig. 4) and k = σ/E, σ being the axial compressive stress. The web is treated as a uniformly compressed and bent plate (2.3) where the load is the reaction r(y) of the intermediate diaphragm (Fig. 5) linearly distributed along the line x = xo. This reaction is represented by the Fourier series (2.2). Hence the general expression (2.16) is obtained for w(x, y). Setting y = h/2 the Eqs. (2.17) are obtained for the edge displacements ξ, ϕ and (2.18) for the edge reactions M1 and P1 of the web. The concentrated reactions P and M of the middle diaphragm (Fig. 7) is represented in the form of a Fourier expansion, Eqs. 2.20 and 2.21, from which the full load of the flange (2.22) is obtained. Substituting (2.22) and (2.17) in (2.1) we obtain, for every n, two equations involving the constants An, Bn, Rm and M. The additional equation (2.28) is obtained from the condition ϕ = 0 for x = x0. Further equations corresponding to the unknowns R, follow from the compatibility condition (2.27) of the displacements of the web and the diaphragm. Two possibilities of making use of this condition are considered. These are the expansion of the identity (2.30) in a Fourier series (2.32) of which the coefficient Zk given by the Eqs. (2.34) should be zero or the satisfaction of the equation (2.27) at one point, thus obtaining equations of the type (2.36).
As a result, an infinite system of homogeneous equations is obtained with the structure shown at Fig. 9. If there is no joint between the middle surface and the web, the unknowns R„ disappear. Successive approximations may be obtained by taking a finite number (n = 2 at least) of terms. In the numerical example the critical value k = 10.02 · 10-3 is obtained while according to Euler's formula k = 10.07 · 10-3. If there are the end diaphragms alone, we obtain, according to [1], k = 2.12 · 10-3. It is seen that the intermediate diaphragm causes the influence of the deform-ability of the cross-section to diminish from 79% to 0.5%, its action thus being very advantageous. The method presented may be used with no essential change for cross-sections of any type and for buckling of any form, any number of intermediate diaphragms and also in the case where the deformability of all the walls is taken into consideration.