Page 190 - Aircraft Stuctures for Engineering Student

174 Structural instability

for a flat plat
In Section 6.3 we saw that the critical load for a column may be determined
experimentally, without actually causing the column to buckle, by means of the
Southwell plot. The critical load for an actual, rectangular, thin plate is found in a
similar manner.
The displacement of an initially curved plate from the zero load position was found
in Section 5.5, to be
cox mrx . nry
wl = xBmnsin-sin-
n h

where

We see that the coefficients Bmn increase with an increase of compressive load intensity
Nx. It follows that when N, approaches the critical value, Nx,CR, the term in the series
corresponding to the buckled shape of the plate becomes the most significant. For a
square plate n = 1 and m = 1 give a minimum value of critical load so that at the
centre of the plate

or, rearranging

Thus, a graph of wl plotted against wl/Nx will have a slope, in the region of the
critical load, equal to Nx,CR.

We distinguished in the introductory remarks to this chapter between primary and
secondary (or local) instability. The latter form of buckling usually occurs in the
flanges and webs of thin-walled columns having an effective slenderness ratio, le/r,
<20. For le/r > 80 this type of column is susceptible to primary instability. In the
intermediate range of le/r between 20 and 80, buckling occurs by a combination of
both primary and secondary modes.
Thin-walled columns are encountered in aircraft structures in the shape of
longitudinal stiffeners, which are normally fabricated by extrusion processes or by
forming from a flat sheet. A variety of cross-sections are employed although each
is usually composed of flat plate elements arranged to form angle, channel, Z- or
‘top hat’ sections, as shown in Fig. 6.17. We see that the plate elements fall into