Civil Engineering Reference
In-Depth Information
columns of any length and to a consideration of beams subjected to axial load and
bending moment.
21.1 E
ULER
T
HEORY FOR
S
LENDER
C
OLUMNS
The first significant contribution to the theory of the buckling of columns was made in
the 18th century by Euler. His classical approach is still valid for long slender columns
possessing a variety of end restraints. Before presenting the theory, however, we shall
investigate the nature of buckling and the difference between theory and practice.
We have seen that if an increasing axial compressive load is applied to a long slender
column there is a value of load at which the column will suddenly bow or buckle in
some unpredetermined direction. This load is patently the buckling load of the col-
umn or something very close to the buckling load. The fact that the column buckles
in a particular direction implies a degree of asymmetry in the plane of the buckle
caused by geometrical and/or material imperfections of the column and its load. The-
oretically, however, in our analysis we stipulate a perfectly straight, homogeneous
column in which the load is applied precisely along the perfectly straight centroidal
axis. Theoretically, therefore, there can be no sudden bowing or buckling, only axial
compression. Thus we require a precise definition of buckling load which may be used
in the analysis of the perfect column.
If the perfect column of Fig. 21.2 is subjected to a compressive load
P
, only shortening
of the column occurs nomatter what the value of
P
. Clearly if
P
were to produce a stress
greater than the yield stress of the material of the column, then material failure would
occur. However, if the column is displaced a small amount by a lateral load,
F
, then, at
values of
P
below the critical or buckling load,
P
CR
, removal of
F
results in a return of
the column to its undisturbed position, indicating a state of stable equilibrium. When
P
P
CR
the displacement does not disappear and the column will, in fact, remain in
any
displaced position so long as the displacement is small. Thus the buckling load,
=
P
Initial
position
Displaced
position
F
P
F
IGURE
21.2
Definition of buckling load of a column
