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In-Depth Information
Material Law:
w
θ
−
V
a
M
a
3
2
/
6
EI
−
/
2
EI
=
2
(4.7c)
/
2
EI
/
EI
v
=
U
s
a
v
s
4.3
Element Matrices, Definitions
4.3.1 Transfer Matrix
Combine the matrices of Eqs. (4.7a), (4.7b), and (4.7c) such that all the variables (forces and
displacements) at
a
are on one side and all variables at
b
are on the other. Then
v
b
=
U
vv
v
a
+
U
v
s
s
a
s
b
=
U
ss
s
a
(4.8a)
or
.
v
b
v
a
U
U
=
vv
v
s
······
······
(4.8b)
.
s
b
s
a
0
U
ss
or
.
−
w
w
3
2
1
−
/
6
EI
−
/
2
EI
01
.
θ
2
θ
/
2
EI
/
EI
=
···
···
······
······
(4.8c)
00
.
V
V
1
0
00
.
M
1
M
b
a
U
i
z
b
=
z
a
(4.8d)
The matrix
U
i
, which is sometimes denoted by
U
i
()
=
U
i
(
−
)
x
b
x
a
,
is referred to as a
transfer
matrix
since it “transfers” the variables
The vector
z
of displacements and forces is called the
state vector
because these variables fully describe
the response or “state” of the beam.
Note from Eqs. (4.7a), (4.7b), and (4.7c) that the partitions of Eqs. (4.8b) and (4.8c) can be
identified with the basic equations for a beam.
w
,
θ
,V,
and
M
from
x
=
x
a
to
x
=
x
b
.
.
.
Geometry
Material
.
.
(
Rigid body
law
U
vv
U
v
s
.
.
displacements
)
···············
·········
······
···
U
i
=
=
(4.9)
.
.
0
.
.
(
Influence of
Equilibrium
0
U
ss
.
.
.
springs
,
foundations
,
(
U
s
v
)
.
etc
.)
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