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TABLE 4.4
Stiffness Matrix for a Beam Element
V
a
M
a
V
b
M
b
V
a
M
a
V
b
M
b
k
11
k
12
k
13
k
14
w
a
θ
a
w
b
θ
b
k
21
k
22
k
23
k
24
=
−
k
31
k
32
k
33
k
34
k
41
k
42
k
43
k
44
P
i
0
p
i
=
k
i
v
i
−
Definitions:
k
11
=
(
e
2
−
η
e
4
)(
e
1
+
ζ
e
3
)
+
λ
/
∇
Sign Convention 2
[
e
3
e
4
]
EI
λ
=
/
k
12
=
[
e
3
(
e
1
−
η
e
3
)
−
e
2
(
e
2
−
η
e
4
)
/
∇
k
EI
]
EI
η
=
k
/
k
s
GA
k
13
=−
(
e
2
−
η
e
4
)
EI
/
∇
k
∗
)/
ζ
=
(
N
−
EI
k
14
=−
e
3
EI
/
∇
ξ
=
EI
/
k
s
GA
k
21
=
k
12
See Table 4.3 for the definitions of
e
i
,i
k
22
={−
(
e
1
−
η
e
3
)
[
e
4
−
ξ(
e
2
+
ζ
e
4
)
]
+
e
2
e
3
}
EI
/
∇
=
1
,
2
,
3
,
4 and
k
23
=
e
3
EI
/
∇=−
k
14
w
0
b
,
θ
b
,V
b
,M
b
0
k
24
=
[
e
4
−
ξ(
e
2
+
ζ
e
4
)
]
EI
/
∇
∇=
e
3
−
(
e
2
−
η
e
4
)
[
e
4
−
ξ(
e
2
+
ζ
e
4
)
]
k
31
=
k
13
EI
is the bending stiffness,
k
s
GA
is
k
32
=
k
23
the shear stiffness
k
33
=
[
(
e
1
+
ζ
e
3
)(
e
2
−
η
e
4
)
+
λ
e
3
e
4
]
EI
/
∇=
k
11
Set 1
/
k
s
GA
=
0 if shear
k
34
={
(
e
1
+
ζ
e
3
)
e
3
+
λ
e
4
[
e
4
−
ξ(
e
2
+
ζ
e
4
)
]
}
EI
/
∇
deformation is not to be
k
41
=
k
14
k
42
=
c
onsidered.
k
24
V
a
=
k
13
w
0
b
+
k
14
θ
0
b
k
43
=
k
34
M
a
=
0
b
0
b
k
23
w
+
k
24
θ
k
44
={
e
2
e
3
−
(
e
1
−
η
e
3
)
[
e
4
−
ξ(
e
2
+
ζ
e
4
)
]
}
EI
/
∇
V
0
b
V
b
+
b
b
=−
k
33
w
+
k
34
θ
=
k
22
M
b
=−
M
b
+
b
b
k
43
w
+
k
44
θ
Finally, Eq. (4.90b) becomes
0
(
p
0
T
4
3
2
−
2
−
z
i
U
i
U
i
))
−
1
P
dx
=
()
(
x
=
−
(2)
120
EI
24
EI
6
which is the result at
x
=
.
If
x
<
,
x
0
(
z
i
U
i
U
i
(τ ))
−
1
P
=
((
x
))
(τ )
d
τ
x
5
120
EI
T
x
4
24
EI
x
2
2
x
3
6
p
0
=
−
−
−
(3)
With the assistance of
U
i
e
A
x
,
the loading vector
z
i
=
can be expressed in the series form
∞
A
j
x
(
j
+
1
)
z
i
=
!
k
!
P
(4)
(
j
+
k
+
1
)
j
=
0
where
k
=
0 for a uniform load,
k
=
1 for a linearly varying load, etc. For example, if
k
=
1
,
then
I
x
2
+
P
A
1
x
2
3!
A
2
x
3
4!
A
3
x
4
5!
z
i
=
+
+
(5)
where use has been made of
A
j
4 and
A
0
=
0 for
j
≥
=
I
,
the unit diagonal matrix. Equation
=
(5) at
x
gives (2).
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