Civil Engineering Reference
In-Depth Information
or
.
/
!
2
x
3
3
L
/
2
x
3
3
L
W
EI
v
C
=−
+
0
"
0
L
/
2
Hence
3
WL
3
8
EI
v
C
=−
as before.
E
XAMPLE
13.14
The cantilever beam shown in Fig. 13.16 tapers along its length
so that the second moment of area of its cross section varies linearly from its value
I
0
at the free end to 2
I
0
at the built-in end. Determine the deflection at the free end
when the cantilever carries a concentrated load
W
.
y
W
A
B
x
I
0
F
IGURE
13.16
Deflection of a cantilever of
tapering section
L
Choosing the origin of axes at the free end B we have, from Eq. (13.10)
x
A
d
v
d
x
A
−
x
B
d
v
d
x
A
M
EI
X
B
−
(
v
A
−
v
B
)
=
x
d
x
(i)
B
in which
I
x
, the second moment of area at any section X, is given by
I
0
1
x
L
I
X
=
+
Also (d
v/
d
x
)
A
=
0,
x
B
=
0,
v
A
=
0 so that Eq. (i) reduces to
L
Mx
EI
0
1
L
v
B
=
d
x
(ii)
x
+
0
The geometry of the
M
/
EI
diagramin this casewill be complicated so that the analytical
approach is most suitable. Therefore since
M
=−
Wx
, Eq. (ii) becomes
L
Wx
2
EI
0
1
v
B
=−
L
d
x
x
+
0
or
L
x
2
WL
EI
0
v
B
=−
x
d
x
(iii)
L
+
0
