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These equations are uncoupled in the linear theory, but are coupled here by
H
d
dx
≈
N
d
dx
=
EA
du
dx
d
dx
(11.26)
This is a nonlinear term that reveals the nonlinear character of the theory of stability. In the
framework of the theory of second order it is assumed that the normal force is known, for
it can be estimated or provided by a linear analysis of the structure
H
d
dx
≈
N
0
d
dx
(11.27)
Upon solution of the equations, which have been linearized by this assumption, the bounds
on the effect of this assumption on the solution accuracy can be checked. If needed, perform
a second analysis with an updated value of the normal force. This becomes an iterative
process, which is typical for nonlinear solutions.
11.2.3 Solution of the Governing Equations
For the case when the coefficients are constant, it is possible to obtain an exact analytical
solution. The system of the first order differential equations of Eq. (11.23) has the solution
(Eq. 4.90)
e
A
x
x
0
e
A
x
z
e
−
A
t
z
(
x
)
=
(
0
)
+
P
(
t
)
dt
=
U
(
x
)
z
(
0
)
+
z
(11.28)
1
1!
A
x
1
2!
A
2
x
2
1
3!
A
3
x
3
1
4!
A
4
x
4
e
A
x
U
(
x
)
=
=
I
+
+
+
+
+···
(11.29)
Similarly, the loading terms can be expanded as
x
I
P
1
1!
A
x
1
2!
A
2
x
2
1
3!
A
3
x
3
1
4!
A
4
x
4
z
(
x
)
=
+
+
+
+
+···
(11.30)
This solution as a matrix series has the advantage that, in general, it can be applied also
in the case of non-constant coefficients of the differential equations. Then, the terms of the
series expansion can be obtained in a straightforward manner, as, for example, in the case
of the stability of curved beams (Wunderlich and Beverungen, 1977).
Alternatively, the solution of the homogeneous part of the higher order relationships of
Eq. (11.25)
EI
d
4
N
0
d
2
w
dx
4
−
w
dx
2
=
0
(11.31)
N
0
2
|
|
2
becomes, with
ε
=
,
EI
Compression,
N
0
=−
P
(11.32)
x
+
x
+
x
w(
x
)
=
C
1
+
C
2
C
3
cos
ε
C
4
sin
ε
Tension,
N
0
=
P
(11.33)
x
+
x
+
x
w(
x
)
=
C
1
+
C
2
C
3
cosh
ε
C
4
sinh
ε
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