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FIGURE 4.15
A general distributed loading
p
z
.
Higher Order Form (Displacement Form) of the Governing Equations
From Chapter 1, Eq. (1.127), the higher order form of the governing differential equations
for an Euler-Bernoulli beam is
dx
=−
θ
d
2
d
w
dx
2
=−
M
EI
(4.82)
d
3
w
dx
3
=−
V
EI
d
4
w
dx
4
=
p
z
EI
d
4
dx
4
Four integrations of the final relationship
(
w/
=
p
z
/
EI
)
lead to
x
EI
d
3
w
dx
3
=−
V
=−
C
1
−
p
z
(τ )
d
τ
0
x
EI
d
2
w
dx
2
=−
M
=−
C
2
−
C
1
x
−
p
z
(τ )
d
τ
0
(4.83)
x
x
2
2
−
d
dx
=−
C
3
EI
−
C
2
x
EI
−
C
1
EI
p
z
(τ )
EI
θ
=−
d
τ
0
x
x
2
2
+
x
3
3!
+
C
4
EI
+
C
3
EI
x
C
2
EI
C
1
EI
p
z
(τ )
EI
w
=
+
d
τ
0
where
p
z
(
x
)
is the transverse distributed loading as shown in Fig. 4.15 and, by definition,
x
x
ξ
d
p
z
(τ )
d
τ
=
p
z
(τ )
d
τ
ξ
0
0
0
and so on. Suppose
x
0 corresponds to the left end, i.e., the “
a
” end, of the beam element.
To rewrite Eq. (4.83) in transfer matrix form, it is necessary to reorganize the constants of
integration
C
1
,C
2
,C
3
,
and
C
4
in terms of the state va
ri
ables at
a
:
=
w
θ
.
a
,
a
,V
a
,
and
M
a
Assume
=
=
.
there is no loading at
x
0 so that the integrals of
p
z
vanish at
x
0
From Eq. (4.83) for
x
=
0
,
C
4
EI
,
C
3
EI
,
w
=
w
=
θ
=
θ
=−
a
x
=
0
a
x
=
0
(4.84)
M
a
=
M
x
=
0
=−
C
2
,
a
=
V
x
=
0
=−
C
1
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