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The transfer matrix is defined by
z
b
=
Uz
a
(11.41)
and is expressed in terms of the values of the analytical solution on both ends of the element
using Eqs. (11.34), (11.36) and (11.37)
N
u
()
···
N
s
()
N
u
(
v
b
···
s
b
0
)
=
c
=
U
···
N
s
(
c
=
Uz
a
(11.42)
0
)
N
z
()
N
z
(
0
)
Therefore, the transition between the mathematical unknowns of the solution and the me-
chanical degrees of freedom at the ends is given by
=
N
z
()
N
−
1
z
U
(
0
)
(11.43)
The transfer matrix, as well as the stiffness matrix, are given explicitly in the following two
sections, accompanied by the respective load vectors for some load cases. In the response
expressions of this section, the nonlinear second order effect is characterized by the specific
parameter
|
N
0
|
ε
=
(11.44)
EI
r
a
nd
the longitudinal strains
N
0
Basically, it contains the
slenderness ratio
/
/
EA
of the
=
√
I
beam column with its radius of gyration
r
/
A
.
11.2.5
Transfer Matrix for a Beam (Theory of Second Order)
The basic relationship for transfer matrices of the
i
th element, as described in Chapter 4, is
given by
z
i
U
i
z
i
z
i
(
x
)
=
(
x
)
(
0
)
+
(
x
)
(11.45)
The state variables with Sig
n
Convention 1 are shown in Fig. 11.13a. The state variable
vector
z
i
and loading vector
z
i
are given by
i
i
u
(
x
)
u
(
x
)
w(
x
)
w(
x
)
θ(
x
)
θ(
x
)
z
i
z
i
(
x
)
=
(
x
)
=
(11.46)
N
(
x
)
N
(
x
)
V
(
x
)
V
(
x
)
M
(
x
)
M
(
x
)
Use of Eq. (11.43) leads to the following transfer matrices, which include second order
effects.
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