Civil Engineering Reference
In-Depth Information
and since d
v
/d
x
is small then
x
1
2
d
v
d
x
1
2
δ
L
δ
+
Hence
1
2
d
x
L
d
v
d
x
1
2
L
=
+
0
giving
L
d
v
d
x
2
1
2
L
+
L
=
d
x
0
Therefore
d
v
d
x
2
L
1
2
L
=
δ =
L
−
d
x
0
Since
L
d
v
d
x
2
1
2
d
x
0
only differs from
d
v
d
x
2
L
1
2
d
x
0
by a term of negligible order, we write
d
v
d
x
2
L
1
2
δ =
d
x
0
giving
L
d
v
d
x
2
P
CR
2
V
=−
d
x
(21.64)
0
The total potential energy of the column in the neutral equilibrium of its buckled state
is therefore
d
v
d
x
2
L
L
M
2
2
EI
d
x
P
CR
2
U
+
V
=
−
d
x
(21.65)
0
0
or, using the alternative form of
U
from Eq. (21.63)
d
2
v
d
x
2
2
L
L
d
v
d
x
2
EI
2
P
CR
2
U
+
V
=
d
x
−
d
x
(21.66)
0
0
