Civil Engineering Reference
In-Depth Information
BUCKLING LOAD FOR A COLUMN WITH FIXED ENDS
In practice, columns usually have their ends restrained against rotation so that they
are, in effect, fixed. Figure 21.5 shows a column having its ends fixed and subjected to
an axial compressive load that has reached the critical value,
P
CR
, so that the column
is in a state of neutral equilibrium. In this case, the ends of the column are subjected
to fixing moments,
M
F
, in addition to axial load. Thus at any section X the bending
moment,
M
, is given by
M
=−
P
CR
v
+
M
F
Substituting for
M
from Eq. (13.3) we have
d
2
v
d
x
2
P
CR
EI
M
F
EI
=−
v
+
(21.6)
Rearranging we obtain
d
2
v
d
x
2
P
CR
EI
M
F
EI
+
v
=
(21.7)
the solution of which is
M
F
P
CR
v
=
C
1
cos
µ
x
+
C
2
sin
µ
x
+
(21.8)
where
P
CR
EI
µ
2
=
When
x
=
0,
v
=
0 so that
C
1
=−
M
F
/
P
CR
. Further
v
=
0at
x
=
L
, hence
M
F
P
CR
M
F
P
CR
0
=−
cos
µ
L
+
C
2
sin
µ
L
+
P
CR
M
F
x
L
EI
0
y
M
F
P
CR
F
IGURE
21.5
Buckling of a slender column with fixed ends
