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summing the result. Thus, for the beam of Fig. 3.10c,
a
1
L
δ
δ
MM
EI
beam 1
MM
EI
beam 2
w
=
+
dx
dx
(2)
0
0
a
1
where moments
δ
M
and
M
remain unchanged.
The slope
θ
of a beam at any location is found by using
M
EI
δ
θ(
1
)
=
Mdx
(3)
where
M
is now the bending moment due to a virtual (unit) moment at the position where
the slope is sought. To find the slope at the free end of the cantilevered beam of Fig. 3.10a,
place a unit moment at
x
δ
=
0 (Fig. 3.10d), and compute the corresponding internal moment
δ
M
to be
δ
M
=−
1
,
so that
L
L
δ
MM
EI
p
0
x
2
2
EI
p
0
L
3
6
EI
θ
=
θ
=
dx
=
dx
=
(4)
x
=
0
0
0
0
EXAMPLE 3.13 Statically Indeterminate Beam with Linearly Varying Loading
Compute the reactions of the statically indeterminate beam of Fig. 3.11.
This beam, with its three reactions, is said to be statically indeterminate to the first degree.
Apply the unit load method to take advantage of the fact that the deflection of the beam
at the right-hand support is zero. The bending moment at any section in terms of the
coordinate
L
3
−
x
from the right end is
M
=
R
L
(
L
−
x
)
−
p
0
(
L
−
x
)
/(
6
L
).
The moment due
FIGURE 3.11
Beam with ramp loading.
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