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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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