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which is the same matrix derived in Chapter 4. The consistent geometric stiffness matrix
becomes
[
w
a
θ
a
w
b
θ
b
]
6
/
5
−
1
/
10
−
6
/
5
−
1
/
10
−
1
/
10
2
/
15
1
/
10
−
1
/
30
1
k
i
geo
=−
N
0
(11.64)
−
6
/
51
/
10
6
/
51
/
10
−
1
/
10
−
1
/
30
1
/
10
2
/
15
which is independent of the material properties of the element.
This geometric stiffness matrix can be obtained from a series expansion of the stiff-
ness matrix of Eq. (11.50). If axial extension is ignored and the displacement vector
v
i
=
b
]
T
[
w
θ
w
θ
is introduced, the stiffness matrix of Eq. (11.50) appears as
a
a
b
(
A
+
B
)
−
D
−
(
A
+
B
)
−
(
A
+
B
)
−
D
]
−
(
A
+
B
)
w
V
a
M
a
2
[2
a
θ
A
A
+
B
B
/
EI
a
w
b
θ
b
=
3
A
+
B
)
−
D
A
+
B
V
b
M
b
/
Symmetric
2
(
A
p
i
k
total
v
i
=
Expand this matrix using the series expressions for
A
,B
,
and
D
of Eq. (11.51). The first
two terms appear as
12
−
6
−
12
−
6
6
/
5
−
1
/
10
−
6
/
5
−
1
/
10
4
6
2
2
/
15
1
/
10
−
1
/
30
EI
1
+···
k
total
=
N
0
3
−
(11.65)
Symmetric
12
6
Symmetric
6
/
51
/
10
4
2
/
15
k
lin
k
i
geo
or
k
total
=
k
lin
+
k
i
geo
+···
Thus, the first term of the series expansion representation of Eq. (11.50) is the usual static
stiffness matrix and the second term is the consistent geometric stiffness matrix.
The expressions for linear and geometric stiffness matrices for beam elements with second
order effects are given in Table 11.3, in which boundary conditions other than the standard
clamped ones are included. For the corresponding load vector, the loading terms consistent
with linear theory, as in Tables 4.2 and 4.4, can be employed.
The accuracy of the approximate series solution relative to the analytical solution of
Section 11.2.6 is of interest. The series representation of the analytical solution may be
truncated at various powers of the parameter
In Fig. 11.15, the solutions for different
levels of truncation are shown together with the solution for a beam (with fixed right
end) according to linear first order theory. For constant
EI
, the exact solution for deflec-
tion
ε.
w
is a cubic polynomial. Although only small differences occur for the deflection
w
,
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