Civil Engineering Reference
In-Depth Information
In designing columns having only one cross-sectional axis of symmetry (e.g. a channel
section) or none at all (i.e. an angle section having unequal legs) the least radius of
gyration is taken in calculating the slenderness ratio. In the latter case the radius of
gyration would be that about one of the principal axes.
Another significant factor in determining the buckling load of a column is the method
of end support. We saw in Section 21.1 that considerable changes in buckling load result
from changes in end conditions. Thus a column with fixed ends has a higher value of
buckling load than if the ends are pinned (cf. Eqs (21.5) and (21.10)). However, we
have seen that by introducing the concept of equivalent length, the buckling loads
of all columns may be referred to that of a pin-ended column no matter what the
end conditions. It follows that Eq. (21.46) may be used for all types of end condition,
provided that the equivalent length,
L
e
, of the column is used. Codes of Practice
list equivalent or 'effective' lengths of columns for a wide variety of end conditions.
Furthermore, although a column buckles naturally in a direction perpendicular to the
axis about which
EI
is least, it is possible that the column may be restrained by external
means in this direction so that buckling can only take place about the other axis.
21.5 S
TABILITY OF
B
EAMS UNDER
T
RANSVERSE AND
A
XIAL
L
OADS
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 geome-
try produced in the loaded beam has an insignificant effect on the loads themselves.
This situation changes drastically when axial loads act simultaneously with the trans-
verse loads. The internal moments, shear forces, stresses and deflections then become
dependent upon the magnitude of the deflections as well as the magnitude of the
external loads. They are also sensitive, as we observed in Section 21.3, to beam imper-
fections such as initial curvature and eccentricity of axial loads. 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
w
and an axial load,
P
, as shown in Fig. 21.13. The bending moment at any
section of the beam is
wx
2
2
=
EI
d
2
v
d
x
2
wLx
2
M
=−
P
v
−
+
(from Eq. 13.3)
giving
d
2
v
d
x
2
P
EI
v
w
2
EI
(
x
2
+
=
−
Lx
)
(21.47)
The standard solution of Eq. (21.47) is
x
2
w
2
P
2
µ
2
v
=
C
1
cos
µ
x
+
C
2
sin
µ
x
+
−
Lx
−
