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And from Eq. (4.116), the transfer matrix is given by
1
−
+
ζ
e
4
−
e
3
/
EI
−
e
4
/
EI
01
−
ζ
e
4
e
2
/
EI
e
3
/
EI
R
−
1
Q
()
R
=
(5)
0
−
ζ
EIe
2
e
1
e
2
0
0
0
1
Some of the terms in (5) can be simplified using
sin
a
sin
a
−
+
ζ
e
4
=−
+
−
=−
=−
e
2
a
a
1
−
ζ
e
3
=
cos
a
=
e
1
,
ζ
EI
=
N
(6)
Then
1
−
e
2
−
e
4
/
EI
−
e
3
/
EI
0
e
1
e
3
/
EI
e
2
/
EI
U
i
R
−
1
Q
=
()
R
=
(7)
00
1
0
−
0
Ne
2
e
2
e
1
It is useful to check to see if this transfer matrix reduces to that for the Euler-Bernoulli
beam with no axial force. If
N
is zero, then
a
2
ζ
=
=
0
.
Also, in the limit as
a
→
0
,e
1
=
=
=
2
/
=
3
/
1
,e
2
,e
3
2
,
and
e
4
3! Then (7) reduces to Eq. (4.86) as desired.
The procedure, as detailed in Example 4.7, readily provides the general transfer matrix
for a beam element that is given in Table 4.3.
A General Stiffness Matrix for Beams
A very general stiffness matrix, including the effect of elastic foundations and shear
deformation, can be obtained by inserting the general transfer matrix components of
Table 4.3 in Eq. (4.16). This leads to the stiffness matrix of Table 4.4.
4.5.2 The Effect of Applied Loading
The
e
ffect on the resp
o
nse of a prescribed loading
P
can be determined using Eq. (4.90b),
i.e.,
z
i
U
i
0
(
It is apparent that this effect can be calculated if the transfer
matrix for the element is known either analytically or numerically.
=
U
i
)
−
1
P
d
τ.
EXAMPLE 4.8 Calculation of Loading Functions
To illustrate the use of Eq. (4.90b) to compute loading functions, suppose an Euler-Bernoulli
beam segment of c
o
nstant cross-section and length
is loaded with a linearly increasing
force described by
p
z
=
/.
As indicated in Eq. (4.93), for a beam with constant
A
,
p
0
x
U
i
))
−
1
U
i
(
(
x
=
(
−
x
).
Then
x
3
6
EI
x
2
2
EI
1
x
−
0
0
0
(
x
2
2
EI
x
EI
01
−
U
i
)
−
1
P
dx
U
i
=
(
−
x
)
P
dx
=
dx
−
p
0
x
/
00
1
0
0
0
0
00
−
x
1
p
0
x
4
6
EI
−
p
0
x
3
2
EI
1
−
=
dx
(1)
−
p
0
x
p
0
x
2
0
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