Civil Engineering Reference
In-Depth Information
P
CR
, is associated with a state of
neutral equilibrium
.For
P
>
P
CR
enforced lateral
displacements increase and the column is unstable.
BUCKLING LOAD FOR A PIN-ENDED COLUMN
Consider the pin-ended column shown in Fig. 21.3. We shall assume that it is in the
displaced state of neutral equilibrium associated with buckling so that the compressive
axial load has reached the value
P
CR
. We also assume that the column has deflected so
that its displacements,
v
, referred to the axes 0
xy
are positive. The bending moment,
M
, at any section X is then given by
M
=−
P
CR
v
so that substituting for
M
from Eq. (13.3) we obtain
d
2
v
d
x
2
P
CR
EI
=−
v
(21.1)
Rearranging we obtain
d
2
v
d
x
2
P
CR
EI
+
v
=
0
(21.2)
The solution of Eq. (21.2) is of standard form and is
v
=
C
1
cos
µ
x
+
C
2
sin
µ
x
(21.3)
in which
C
1
and
C
2
are arbitrary constants and
µ
2
=
P
CR
/
EI
. The boundary conditions
for this particular case are
v
=
0at
x
=
0 and
x
=
L
. The first of these gives
C
1
=
0 while
from the second we have
0
=
C
2
sin
µ
L
x
P
CR
EI
L
y
0
F
IGURE
21.3
Determination of buckling load for a
pin-ended column
P
CR
