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FIGURE 5.5
Beam with ramp loading (Examples 5.2 and 5.6).
The state variables can be computed between nodes by adjusting the
x
coordinate (
) in the
transfer matrix for that element.
Procedures to improve the computational efficiency and the numerical stability are
treated in Section 5.1.5.
EXAMPLE 5.2 Beam with Linearly Varying Loading
The fixed-simply supported uniform beam of Fig. 5.5 has been treated frequently in this
topic. Suppose it is modeled with two elements and the response is to be found using the
transfer matrix method.
The transfer matrices for each element can be taken from Chapter 4, Table 4.3 as
. (
3
2
4
1
−
−
/
6
EI
−
/
2
EI
4
p
0
+
p
0
/
2
)
/
120
EI
.
−
(
0
1
2
/
2
EI
/
EI
3
p
0
+
p
0
/
2
)
3
/
24
EI
.
0
0
1
0
−
(
p
0
+
p
0
/
2
)/
2
U
1
=
(1)
.
−
(
+
/
)
2
/
0
0
1
2
p
0
p
0
2
6
...
...
......
......
.
..................
.
0
0
0
0
1
L
2
=
.
3
2
4
1
−
−
/
6
EI
−
/
2
EI
p
0
/
60
EI
.
−
2
3
01
/
2
EI
/
EI
p
0
/
16
EI
.
00
1
0
−
p
0
/
4
U
2
=
(2)
.
00
1
−
p
0
2
/
6
... ...
......
......
.
............
.
00
0
0
1
L
2
=
The overall transfer matrix
U
would be
U
2
U
1
z
a
=
z
x
=
L
=
z
c
=
Uz
a
(3)
The initial parameter vector
z
a
is determined by applying boundary conditions to (3). As
is always the case, two of the four initial parameters are evaluated by observation. For the
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