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element indices have been neglected. The element contribution of Eq. (4.61) more properly
should appear as
v
iT
p
i
p
i
0
v
iT
k
i
v
i
p
i
0
δ
(
−
)
=
δ
(
−
)
(4.62)
We wish to evaluate the stiffness matrix
k
i
=
[1
xx
2
x
3
]of
of Eq. (4.57). Begin with
N
u
Eq. (4.41). Then
d
2
N
u
(
x
)
N
u
(
B
u
(
x
)
=
x
)
=
=
[0026
x
]
(4.63)
dx
2
For constant
EI
, the integration in the expression for
k
of Eq. (4.56) leads to
00 0
0
EI
0
00 0
0
B
u
EI
B
u
dx
=
dx
(4.64)
00 4
12
x
0
36
x
2
0012
x
00 0
0
00 0
0
=
EI
2
00 4
6
2
3
006
12
Finally, use
G
of Eq. (4.45) in Eq. (4.57) to obtain
12
−
6
−
12
−
6
EI
2
2
−
6
4
6
2
k
i
=
(4.65)
−
12
6
12
6
3
2
2
−
6
2
6
4
Stiffness Matrix Based on Normalized Coordinate
To calculate the stiffness matrix for an Euler-Bernoulli beam element using the normalized
coordinate
ξ
=
/
x
, begin by defining the assumed displacement (Eq. 4.47a)
Gv
e
w(ξ)
=
N
u
(ξ )
(4.66)
with
G
given in Eq. (4.47a), and
N
u
=
ξξ
2
ξ
3
]
[1
(4.67a)
v
i
=
w
a
θ
a
w
b
θ
b
]
iT
[
(4.67b)
The corresponding force vector is
p
i
=
M
b
]
iT
[
V
a
M
a
V
b
(4.67c)
We have chosen to use the same
v
i
and
p
i
that were employed in evaluating
k
i
of Eq.
(4.57). The stiffness matrix
k
i
as a function of
x
is calculated using
b
k
i
B
T
B
T
=
(
x
)
EI
B
(
x
)
dx
=
(
x
)
EI
B
(
x
)
dx
a
0
G
T
0
B
u
(
=
x
)
EI
B
u
(
x
)
dx
G
(4.68)
with
G
from Eq. (4.45). In terms of the normalized coordinate,
dx
=
d
ξ
and
d
2
dx
2
N
u
(ξ )
=
d
2
N
u
(ξ )
1
N
u
(ξ )
=
B
u
(ξ )
=
=
2
[0026
ξ
]
(4.69)
2
d
ξ
2
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