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a factor for the fundamental state corresponding to a certain combination of the loading.
Sometimes
is referred to as the
load factor
. It can, for example, represent the influence of
the fundamental state in the form
λ
2
2
N
0
λ
=
ε
=
/
EI
.
cr
(lowest eigenvalue of the system) can be obtained relatively simply
by the so-called inverse vector-iteration method (See, for example, Zum uhl, 1963)
The critical load
λ
K
lin
V
i
=−
λ
i
−
1
K
geo
V
i
−
1
(11.68)
λ
which always converges toward the lowest eigenvalue
min
. The iteration is started by an
λ
arbitrary vector
V
0
and
0
and should be scaled with respect to a constant length for each
step, e.g.,
|
V
i
|=
1:
V
i
V
i
=
V
i
/
a
with
a
=
V
i
(11.69)
The columns of Examples 11.6 and 11.7 will be considered again to demonstrate the pro-
cedure. It can be observed that the buckling loads obtained will be higher than the exact
value. This is due to the approximate nature of the geometric matrix. The results may be
improved by subdividing the model of the structures into more elements.
EXAMPLE 11.10 Column Buckling
For the hinged-hinged column of Example 11.6 (Fig. 11.27) the buckling load was found as
the lowest root of
A
2
B
2
−
=
0
(1)
This led to
2
EI
L
2
=
ε
P
cr
(2)
2
2
870. If
A
and
B
are expanded in terms of
2
, then (Eq. 11.51)
with
ε
=
π
=
9
.
ε
4
2
2
2
2
2
2
15
ε
1
30
ε
A
2
B
2
−
=
−
−
+
=
0
(3)
2
where only terms up to
ε
are retained. Equation (3) becomes
1
60
ε
6
5
ε
4
2
−
+
12
=
0
(4)
2
giving
464. The corresponding critical load is, as expected, higher than
the exact value of (2). For design, note that this approximation is on the unsafe side.
For the fixed-hinged column of Example 11.7 (Fig. 11.28), the critical load is obtained
from
ε
=
12 and
ε
1
=
3
.
A
=
0
(5)
2
2
or
leading to (2) with
ε
=
2
.
05
π
ε
1
=
4
.
493. If the expansions of Eq. (11.51) are introduced
into (5), we find
2
15
ε
A
=
2
4
−
=
0
(6)
2
2
when terms up to
ε
are retained. This leads to
ε
=
30 or the lowest root of
ε
1
=
5
.
477. If
4
terms up to
ε
are retained
2
15
ε
11
6300
ε
A
=
2
4
4
−
−
=
0
(7)
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