Civil Engineering Reference
In-Depth Information
x
Load intensity at section X is
w
0
L
y
X
w
0
B
A
x
EI
F
IGURE
13.8
Deflection
of a simply supported
beam carrying a
triangularly distributed
load
x
R
A
R
B
L
Resolution of vertical forces then gives
w
0
L
3
R
B
=
The bending moment,
M
, at any section X, a distance
x
from A is
2
w
0
L
x
x
1
x
M
=
R
A
x
−
3
or
w
0
6
L
(
L
2
x
x
3
)
M
=
−
(i)
Substituting for
M
in Eq. (13.3) we obtain
EI
d
2
v
d
x
2
w
0
6
L
(
L
2
x
x
3
)
=
−
(ii)
which, when integrated, becomes
L
2
x
2
2
x
4
4
EI
d
v
d
x
w
0
6
L
=
−
+
C
1
(iii)
Integrating Eq. (iii) we have
L
2
x
3
6
x
5
20
w
0
6
L
EI
v
=
−
+
C
1
x
+
C
2
(iv)
The deflection
v
=
0at
x
=
0 and
x
=
L
. From the first of these conditions we obtain
C
2
=
0, while from the second
L
5
6
L
5
20
w
0
6
L
0
=
−
+
C
1
L
which gives
7
w
0
L
4
360
C
1
=−
