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FIGURE 11.9
Simply supported column with eccentrically applied
axial force.
These are closed form solutions composed of functions that are defined analytically. It
should be kept in mind, however, that these functions are also defined through a power
series expansion and are computed as such in practical calculations on the computer.
The approach using the matrix series appears to be more direct and more generally
applicable.
EXAMPLE 11.3 Stability of a Column with Eccentrically Applied Axial Load
To study the stability of a beam, consider the simply supported column of Fig. 11.9 with an
eccentrically applied axial force.
The boundary conditions are
w(
0
)
=
0
,
M
(
0
)
=
Pe,
w(
L
)
=
0
,
M
(
L
)
=
Pe
(1)
Apply
th
ese boundary conditions to the solution of Eq. (11.32), with
N
0
=−
P,
so that
2
L
2
P
ε
=
/
EI
.
10
1
0
C
1
C
2
C
3
C
4
0
0
w(
0
)
=
0
x
L
w(
L
)
=
0
11
c
(
L
)
s
(
L
)
(
)
=
ε
s
x
sin
=
x
L
(
)
=
2
L
2
2
EI
L
2
(
)
=
ε
M
0
Pe
00
EI
ε
/
0
e
ε
/
c
x
cos
M
(
L
)
=
Pe
2
c
L
2
2
s
L
2
2
EI
L
2
00
EI
ε
(
L
)/
EI
ε
(
L
)/
e
ε
/
(2)
The vector on the right-hand side of this relationship contains the homogenous terms that
would be zero for a pure column (with no eccentricity). The constants
C
1
,C
2
,C
3
,
and
C
4
,
found by solving (2) are substituted into Eq. (11.32) giving
e
1
1
−
cos
ε
x
L
+
x
L
−
w(
x
)
=
sin
ε
cos
ε
(3)
sin
ε
The maximum value of the deflection
e
1
1
cos
w(
L
/
2
)
=
2
−
(4)
ε/
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