Information Technology Reference
In-Depth Information
In more general terms for an element of length
,
extending from
x
=
x
a
to
x
=
x
b
,
U
i
z
a
x
b
x
a
(
U
i
)
−
1
P
d
U
i
z
a
z
i
z
b
=
+
τ
=
+
(4.90a)
where
U
i
z
i
z
i
b
=
U
i
(τ ))
−
1
P
=
0
(
(τ )
d
τ
(4.90b)
and, as indicated above for a constant coefficient matrix
A
,
U
i
U
i
e
A
(
x
b
−
x
a
)
=
()
=
(4.90c)
−
=
.
with
x
b
x
a
The exponential relation of Eq. (4.90c) can be expanded in the series
∞
A
2
2
A
s
s
A
1!
+
U
i
e
A
=
=
I
+
+···=
(4.91)
2!
s
!
=
s
0
where
I
is the identity matrix, a diagonal matrix with diagonal values of unity. Such an
expansion lends itself well for numerical calculations for more complicated members than
simple beams, as it is often possible to control the error. From Eq. (4.90c), the exponential
solution leads to a loading term of the form
e
A
(
x
b
−
x
a
)
x
b
x
a
z
i
e
−
A
(τ
−
x
a
)
P
d
=
τ
(4.92)
U
i
))
−
1
e
−
A
x
It follows from
(
(
x
=
that for constant
A
U
i
))
−
1
U
i
(
(
x
=
(
−
x
)
(4.93)
a result that can be useful when finding the loading vector
z
i
.
EXAMPLE 4.6 Transfer Matrix for an Euler-Bernoulli Beam
For the Euler-Bernoulli beam, with the governing equations of Eq. (4.88), the transfer matrix
is obtained from Eq. (4.91) using
0
−
10
0
0
EI
000 0
001 0
001
/
A
=
00
0
−
1
/
EI
001
/
EI
0
A
2
=
AA
=
(1)
00
0
0
00
0
0
00
−
1
/
EI
0
0
0
0
0
A
3
AA
2
=
=
0
0
0
0
0
0
0
0
Continued multiplication of
A
shows that
A
4
0 and that
A
s
0 for any
s
greater than
3. The termination of the expansion is to be expected since the analytical solution for this
=
=
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