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4.3.2
Stiffness Matrix
A stiffness matrix for the beam element relates all of the displacements at
a
and
b
, i.e.,
w
b
, to all the forces, i.e.,
V
a
,M
a
,V
b
,
and
M
b
. The stiffness matrix
k
i
a
,
θ
a
,
w
b
,
and
θ
for
element
i
is defined as
p
i
k
i
v
i
=
(4.10)
where
V
a
M
a
V
b
M
b
w
a
θ
a
w
b
θ
b
p
a
p
b
v
a
v
b
p
i
v
i
=
=
=
=
k
11
k
12
k
13
k
14
k
aa
k
ab
k
21
k
22
k
23
k
24
k
i
=
=
k
ba
k
bb
k
31
k
32
k
33
k
34
k
41
k
42
k
43
k
44
The stiffness matrix is defined using
p
of Sign Convention 2, Eq. (4.2). The stiffness matrix
is an important building block for the analysis of structural systems. The fundamental
relations of Eqs. (4.7a), (4.7b), and (4.7c) were placed together in a special manner to form
the transfer matrix. The same fundamental relations can be reorganized to form the stiffness
matrix. This is to be expected since both the transfer and stiffness matrices are relationships
between the same eight variables
θ
b
,V
a
,M
a
,V
b
,
and
M
b
. Of course, there are
numerous other methods for finding the stiffness matrix, some of which were treated in
Chapter 3, while others will be considered in this section.
Consider first the arrangement of the fundamental relations of Eqs. (4.7a), (4.7b), and
(4.7c) into a stiffness matrix. From Eq. (4.8a), written in terms of Sign Convention 2 [replace
s
by
p
and use Eq. (4.3)],
w
a
,
θ
a
,
w
b
,
p
b
=
U
pp
p
a
,
v
b
=
U
vv
v
a
+
U
v
p
p
a
where
U
pp
=−
U
ss
and
U
v
p
=−
U
v
s
.
It follows that
U
−
1
v
U
−
1
v
p
a
=
p
v
b
−
p
U
v
a
vv
U
pp
U
−
1
v
U
pp
U
−
1
v
p
b
=
U
pp
p
a
=
p
v
b
−
p
U
v
a
vv
or
=
.
p
a
v
a
U
−
1
v
U
−
1
v
−
p
U
vv
p
·········
·········
(4.11)
···
p
b
···
v
b
.
U
pp
U
−
1
v
U
pp
U
−
1
v
−
p
U
vv
p
k
i
where
U
−
1
v
is given by (Sign Convention 2)
p
−
3
2
12
EI
/
−
6
EI
/
U
−
1
v
=
p
2
6
EI
/
2
EI
/
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