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From Eq. (11.23), with
N
=
P
EI
d
3
w
dx
3
−
P
d
dx
V
=−
(2)
where
N
is compressive. From Eqs. (1) and (2) at
x
=
L
,
EI
d
3
w
dx
3
+
P
d
dx
+
P
L
1
L
=
0
(3)
x
=
w(
)
=
w
(
)
=
(
)
=
This and the relations
0
0
0
,M
L
0 are the boundary conditions for the
column.
From Eq. (11.32), for 0
≤
x
≤
L
,
x
L
+
x
L
+
x
L
w
=
C
1
+
C
2
C
3
cos
ε
C
4
sin
ε
C
3
L
x
L
+
C
4
L
x
L
w
=
C
2
−
sin
ε
cos
ε
L
2
P
EI
C
3
L
2
C
4
L
2
2
ε
=
(4)
x
L
−
x
L
w
=−
cos
ε
sin
ε
C
3
L
2
C
4
L
3
x
L
−
x
L
w
=
sin
ε
cos
ε
w
(
w
(
The boundary conditions
w(
0
)
=
0
,
0
)
=
0, and
L
)
=
0 and (3) imposed on (4)
become, with the abbreviations
s
=
sin
ε
,c
=
cos
ε
,
=
1
0
1
0
C
1
C
2
C
3
C
4
0
1
0
ε
0
(5)
0
0
−
c
−
s
11
+
L
1
/
Lc s
The critical axial force is obtained by setting the determinant of these relations equal to
zero. Thus,
1
0
1
0
=
ε
0
1
0
=−
L
(
det
0
1
+
L
1
/
L
)
c
+
s
(6)
0
0
−
c
−
s
11
+
L
1
/
Lc s
or
tan
ε
=
ε(
1
+
L
1
/
L
)
(7)
This characteristic equation is plotted in Fig. 11.31, where the straight lines projecting ra-
dially from 0 represent the right-hand side of (7). The curved lines in Fig. 11.31 are plots of
tan
0 correspond to segments of length
L
1
lying to the left of the
moment release. For particular ranges of values
L
1
, the buckling loads can be determined.
ε.
The cases for which
L
1
<
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