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Unit Load Method
A simple technique for finding
w
and
θ
is the unit load method of Chapter 3. To compute
the displacement (
w
V
) and slope (
θ
V
) due to
V
b
at the right end, apply a unit force
downward (for obtaining
θ
V
), and use the moment
diagrams of Figs. 4.3c and e. Assume
EI
is constant. It follows from Chapter 3, Eq. (3.17)
that
w
V
) and a unit moment (for obtaining
(
−
)
=
3
V
b
3
EI
V
b
x
w
=
(
−
)
x
dx
V
EI
0
2
V
b
2
EI
V
b
(
−
x
)
=−
θ
V
=−
1
dx
EI
0
For the displacement (
θ
M
) due to
M
b
at the right end, use the moment
diagrams of Figs. 4.3d and e. From Eq. (3.17)
w
M
) and slope (
0
−
M
b
EI
(
−
=−
2
M
b
2
EI
w
M
=
x
)
dx
M
b
EI
1
dx
=
M
b
EI
θ
M
=
0
Then
3
V
b
2
M
b
2
EI
w
=
w
V
+
w
M
=
3
EI
−
2
V
b
=−
2
EI
+
M
b
EI
θ
=
θ
+
θ
V
M
Substitute the equilibrium conditions,
V
b
=
V
a
,M
b
=
M
a
+
V
a
,
into these relations to
w
θ
obtain expressions for
and
as functions of
V
a
and
M
a
.
3
6
EI
V
a
2
2
EI
M
a
w
=−
−
(4.6)
2
2
EI
V
a
EI
M
a
θ
=
+
4.2.4 Summary
In matrix notation, the fundamental relations using Sign Convention 1 appear as
Equilibrium:
V
b
M
b
10
V
a
M
a
=
1
(4.7a)
s
b
=
U
ss
s
a
Geometry:
w
1
w
w
θ
−
01
b
a
=
+
θ
θ
b
a
(4.7b)
v
b
=
U
vv
v
a
+
v
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