Page 178 - Aircraft Stuctures for Engineering Student

162 Structural instability

well known Southwell plot for the experimental determination of the elastic buckling
load of an imperfect column.
Timoshenko' also showed that Eq. (6.27) may be used for a perfectly straight
column with small eccentricities of column load.

Stresses and deflections in a linearly elastic beam subjected to transverse loads as
predicted by simple beam theory, are directly proportional to the applied loads.
This relationship is valid if the deflections are small such that the slight change in
geometry produced in the loaded beam has an insignificant effect on the loads
themselves. This situation changes drastically when axial loads act simultaneously
with the transverse loads. The internal moments, shear forces, stresses and deflections
then become dependent upon the magnitude of the deflections as well as the magni-
tude of the external loads. They are also sensitive, as we observed in the previous
section, to beam imperfections such as initial curvature and eccentricity of axial
load. Beams supporting both axial and transverse loads are sometimes known as
beam-columns or simply as transversely loaded columns.
We consider first the case of a pin-ended beam carrying a uniformly distributed
load of intensity MI per unit length and an axial load P as shown in Fig. 6.9. The
bending moment at any section of the beam is

wlz wz 2 d2u
= pu + - - -EI - (see Section 9.1)
=
-
2 2 dz2
giving
P
d2u
-+-u=-(z w 2 -1z) (6.29)
dz2 EI 2EI
The standard solution of Eq. (6.29) is

I w/ u n i t length I
I L I

Fig. 6.9 Bending of a uniformly loaded beam-column.