Wpływ zmniejszania się mimośrodu na ugięcia prętów mimośrodowo ściskanych

The influence of decreasing eccentricity on the deflections of a bar subjected to eccentric compression

In all investigations concerning the computation of finite deflections of eccentrically compressed bars it has been assumed hitherto that the eccentricity of the force remains constant («normal to the force», Fig. la) during the deformation process. In addition to this type of load, which can be realized, for instance, in the somewhat unusual way shown at Fig. 2 or, more .commonly, with the axial force and the moment acting independently (the values' increasing proportionally), various other types of eccentric load are possible. Above all, there is the case shown at Fig. 1b, where the bar representing the eccentricity remains always normal to the tangent to the loaded bar («the decreasing eccentricity»).

The object of this paper is to compare the deflections for the two types of load. In particular, since the analysis for constant eccentricity is simp­ler, and the decreasing eccentricity seems to be more often encountered in practice, the differences (absolute and relative) between these two cases are investigated. The cases in which these differences can be dis­regarded are indicated.

The notations used are those of the previous paper, [18], by the present author. Using the exact equation of the deflection curve (2.1), the Eqs. (2.20) are obtained for the maximum deflection in the case of decreasing eccentricity and (2.21) in the case of constant eccentricity. These are equations of the type f(ϑ, Θ, m) = 0 , the unknown being ϑ = δ/l, where l denotes the length of the bar and Θ, and m the dimensionless eccentricity and dimensionless axial force, respectively.

In Art. 3 these equations are expressed in a parametric form which makes possible a computation of the sets of variables ϑ, Θ and m, corresponding each other. Further calculations are performed graphically and numerically. Thus, the diagram of the curve Θ = const in the ϑ m plane is constructed (Fig. 3), together with the diagram of the curve ϑ = const in the plane Θ m (Fig. 4). On the basis of Fig. 4, are drawn diagrams of the absolute and relative (percentage) deflection differences between the two types of load (Figs. 5 and 6).

In conclusion, it can be assumed that the differences between the two types of load can be disregarded when the eccentricity of the force is of the order of a thousandth part of the bar length.

A four-figure table of deflections of a bar under buckling (axial compression), calculated from the accurate equation (4.7), is presented too. This case can be treated as the limit case for the problem under consideration, the difference between the two types of load being zero.