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w
∗
,
θ
∗
,M
∗
,
and
V
∗
are the responses of an infinite beam at point
x
due to a unit force
where
at point
ξ
and
1
when
x
>ξ
sgn
(
x
−
ξ)
=
−
1
when
x
<ξ
w
∗
has been omitted.
An arbitrary integration constant in the expression for
can be interchanged, as Maxwell's reciprocal theorem [Chapter 3,
Eq. (3.38)] asserts that for linear problems the responses at point
x
due to a unit force at
point
Note that
x
and
ξ
ξ
are equal to the responses at point
ξ
due to a unit force at
x
. It should be noted that
when the positions of
x
and
ξ
are interchanged,
x
becomes the source point and
ξ
becomes
θ
∗
,M
∗
,
and
V
∗
should
the integration variable. Then, the derivatives in the expressions for
be taken with respect to
=
δw
|
ξ
=
L
,
etc.
Equation (9.4) will then become the expression for the displacement at point
x
.
With Eq. (9.5), the integral on the left-hand side of Eq. (9.4) becomes
L
0
δw
ξ
and
dx
of Eq. (9.4) should be changed to
d
ξ
,
δw
L
L
0
δ(ξ
i
v
EI
w
dx
=
,x
)w(
x
)
dx
=
w(ξ)
so that Eq. (9.4), with the help of Eq. (9.7), can be transformed to
L
0
w
∗
(ξ
V
0
w
M
0
θ
V
L
w
M
L
θ
w(ξ)
=
,x
)
p
z
(
x
)
dx
+
+
−
−
0
0
L
L
−
w
0
V
0
−
θ
0
M
0
+
w
L
V
L
+
θ
L
M
L
(9.8)
with
V
∗
(ξ
V
0
,
0
M
∗
(ξ
V
L
=
V
∗
(ξ
δ
V
0
=
δ
V
|
x
=
0
=
,
0
)
=
=
,
0
)
,
,L
)
,
etc.
The starred terms in Eq. (9.8) can be written as, for example,
1
2
[sgn
1
2
V
0
=
V
∗
(ξ
,
0
)
=−
(
0
−
ξ)
]
=
1
2
[sgn
−
ξ)
=−
2
M
0
=
M
∗
(ξ
,
0
)
=−
(
0
−
ξ)
]
(
0
1
12
EI
[sgn
1
12
EI
(
w
L
=
w
∗
(ξ
3
3
,L
)
=
(
L
−
ξ)
]
(
L
−
ξ)
=
L
−
ξ)
Substitute these relationships into Eq. (9.8) to obtain
L
0
w
∗
(ξ
1
2
w
0
−
2
θ
0
+
1
2
w
L
+
(
L
−
ξ)
2
w(ξ)
=
,x
)
p
z
(
x
)
dx
+
θ
L
3
12
EI
V
0
2
4
EI
M
0
3
2
ξ
ξ
+
(
L
−
ξ)
−
(
L
−
ξ)
−
−
V
L
M
L
(9.9)
12
EI
4
EI
Equation (9.9) is the expression for the deflection at point
ξ
along the beam axis. It can be
seen that the deflection is a function of
ξ
only and
x
is not involved. The slope should be
expressed as
θ(ξ)
=−
dw
/
d
ξ.
The derivative of
w
with respect to
ξ
is given by
dw
d
w
(ξ )
=
ξ
=−
θ(ξ)
2
4
EI
V
0
2
1
2
θ
1
2
θ
ξ
ξ
2
EI
M
0
−
(
L
−
ξ)
+
(
L
−
ξ)
2
EI
=−
−
−
−
V
L
M
L
0
L
4
EI
L
0
w
∗
(ξ
+
,x
)
p
z
(
x
)
dx
(9.10)
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