Civil Engineering Reference
In-Depth Information
(21.10), (21.15) and (21.21) we see that the equivalent lengths of the various types of
column are
both ends pinned
L
e
=
1
.
0
L
both ends fixed
L
e
=
0
.
5
L
one end fixed and one free
L
e
=
2
.
0
L
one end fixed and one pinned
L
e
=
0
.
7
L
21.2 L
IMITATIONS OF
T
HE
E
ULER
T
HEORY
For a column of cross-sectional area
A
the critical stress,
σ
CR
, is, from Eq. (21.23)
π
2
EI
AL
e
P
CR
A
σ
CR
=
=
(21.24)
The second moment of area,
I
, of the cross section is equal to
Ar
2
where
r
is the
radius
of gyration
of the cross section. Thus we may write Eq. (21.24) as
π
2
E
(
L
e
/
r
)
2
σ
CR
=
(21.25)
Therefore for a column of a given material, the critical or buckling stress is inversely
proportional to the parameter (
L
e
/
r
)
2
.
L
e
/
r
is an expression of the proportions of the
length and cross-sectional dimensions of the column and is known as its
slenderness
ratio
. Clearly if the column is long and slender
L
e
/
r
is large and
σ
CR
is small; conversely,
for a short column having a comparatively large area of cross section,
L
e
/
r
is small
and
σ
CR
is high. A graph of
σ
CR
against
L
e
/
r
for a particular material has the form
shown in Fig. 21.9. For values of
L
e
/
r
less than some particular value, which depends
upon the material, a column will fail in compression rather than by buckling so that
σ
CR
as predicted by the Euler theory is no longer valid. Thus in Fig. 21.9, the actual
failure stress follows the dotted curve rather than the full line.
s
CR
Euler theory
Actual failure stress
F
IGURE
21.9
Variation of critical
stress with slenderness ratio
L
e
/r
