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These transfer matrices were also derived in Chapter 4 in a different notation and are
tabulated in a rather general form in Table 4.3. For the corresponding load vectors, see also
Table 4.3.
11.2.6
Stiffness Matrix for a Beam (Theory of Second Order)
The basic relationship for stiffness matrices of the
i
th element, as described in Chapter 4, is
given by
p
i
0
The definition of the state variables using Sign Convention 2 is shown in Fig. 11.13b referred
to the undeformed state. They form the vector of the force variables
p
i
and the displacement
variables
v
i
p
i
k
i
v
i
=
−
for element
i
.
i
i
N
a
V
a
M
a
N
b
V
b
M
b
u
a
w
a
θ
a
u
b
w
b
θ
b
p
i
v
i
=
=
(11.49)
Any number of methods can be employed to obtain the stiffness matrix
k
i
for a beam element
subjected to an axial force. For example, Chapter 4, Eq. (4.11) can be used to convert the
transfer matrices of Eqs. (11.47) and (11.48) into stiffness matrices. Also the development of
Eq. (11.40) can be utilized to obtain the following stiffness matrices which include second
order effects:
Bar in Compression
(
N
0
<
0)
N
0
EI
ε
=
.
−
N
a
u
a
EA
/
0
0
EA
/
0
0
2
.
V
a
w
a
[2
(
A
+
B
)
−
D
]
EI
/
3
−
(
A
+
B
)
EI
/
0
−
[2
(
A
+
B
)
−
D
]
EI
/
3
−
(
A
+
B
)
EI
/
2
.
M
a
θ
a
A
EI
A
+
B
)
2
B
EI
=
/
0
(
EI
/
/
......
.
...
.........
.........
N
b
u
b
.
EA
/
0
0
V
b
w
b
A
+
B
)
−
D
]
EI
3
A
+
B
)
2
Symmetric
[2
(
/
(
EI
/
M
b
θ
b
A
EI
/
p
i
k
i
v
i
(11.50)
=
where the parameters
A
,B
,D
are (Chwalla, 1959)
ε(
sin
ε
−
ε
cos
ε)
2
15
ε
11
6300
ε
2
4
A
=
A
=
4
−
−
−···
2
(
1
−
cos
ε)
−
ε
sin
ε
ε(ε
−
sin
ε)
1
30
ε
13
12 600
ε
B
=
B
=
2
4
2
+
+
−···
2
(
1
−
cos
ε)
−
ε
sin
ε
ε
2
(
−
ε)
1
cos
1
10
ε
1
1400
ε
2
4
A
+
B
=
A
+
B
=
−
−
−···
6
2
(
1
−
cos
ε)
−
ε
sin
ε
|
|
EI
N
D
=
ε
2
2
=
ε
3
sin
ε
6
5
ε
1
700
ε
2
4
(
A
+
B
)
−
D
=
(
A
+
B
)
−
D
=
−
−
−···
(11.51)
2
2
12
(
−
ε)
−
ε
ε
2
1
cos
sin
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