Page 176 - Aircraft Stuctures for Engineering Student

160 Structural instability

Extensive experiments carried out on aluminium alloy columns by the aircraft
industry in the 1940s showed that the actual buckling load was approximately
equal to the tangent modulus load. Shanley (1947) explained that for columns with
small imperfections, an increase of axial load and bending occur simultaneously.
He then showed analytically that after the tangent modulus load is reached, the
strain on the concave side of the column increases rapidly while that on the convex
side decreases slowly. The large deflection corresponding to the rapid strain increase
on the concave side, which occurs soon after the tangent modulus load is passed,
means that it is only possible to exceed the tangent modulus load by a small
amount. It follows that the buckling load of columns is given most accurately for
practical purposes by the tangent modulus theory.
Empirical formulae have been used extensively to predict buckling loads, although
in view of the close agreement between experiment and the tangent modulus theory
they would appear unnecessary. Several formulae are in use; for example, the
Rankine, Straight-line and Johnson's parabolic formulae are given in many books
on elastic stability'.

Obviously it is impossible in practice to obtain a perfectly straight homogeneous
column and to ensure that it is exactly axially loaded. An actual column may be
bent with some eccentricity of load. Such imperfections influence to a large degree
the behaviour of the column which, unlike the perfect column, begins to bend
immediately the axial load is applied.
Let us suppose that a column, initially bent, is subjected to an increasing axial load
P as shown in Fig. 6.8. In this case the bending moment at any point is proportional
to the change in curvature of the column from its initial bent position. Thus

d2v d2vo
EI- - EI- == -Pv (6.22)
dz2 dz2
which, on rearranging, becomes

d2v d2vo
-+Av=- (6.23)
dz2 dz2

Z

Fig. 6.8 Initially bent column.