Civil Engineering Reference
In-Depth Information
X
y
B
M
0
A
C
x
EI
M
0
L
M
0
L
F
IGURE
13.12
Deflection of a simply
supported beam carrying
a point moment
(Ex. 13.10)
R
A
R
C
b
L
E
XAMPLE
13.10
Determine the deflected shape of the beam shown in Fig. 13.12.
In this problem an external moment
M
0
is applied to the beam at B. The support
reactions are found in the normal way and are
M
0
L
M
0
L
R
A
=−
(downwards)
R
C
=
(upwards)
The bending moment at any section X between B and C is then given by
M
=
R
A
x
+
M
0
(i)
Equation (i) is valid only for the region BC and clearly does not contain a singularity
function which would cause
M
0
to vanish for
x
≤
b
. We overcome this difficulty by
writing
b
]
0
b
]
0
M
=
R
A
x
+
M
0
[
x
−
(Note: [
x
−
=
1)
(ii)
Equation (ii) has the same value as Eq. (i) but is now applicable to all sections of the
beam since [
x
b
]
0
disappears when
x
−
≤
b
. Substituting for
M
from Eq. (ii) in Eq.
(13.3) we obtain
EI
d
2
v
d
x
2
b
]
0
=
R
A
x
+
M
0
[
x
−
(iii)
Integration of Eq. (iii) yields
x
2
2
EI
d
v
d
x
=
R
A
+
M
0
[
x
−
b
]
+
C
1
(iv)
and
x
3
6
+
M
0
2
b
]
2
EI
v
=
R
A
[
x
−
+
C
1
x
+
C
2
(v)
where
C
1
and
C
2
are arbitrary constants. The boundary conditions are
v
=
0 when
x
=
0 and
x
=
L
. From the first of these we have
C
2
=
0 while the second gives
L
3
6
+
M
0
L
M
0
2
b
]
2
0
=−
[
L
−
+
C
1
L
