Civil Engineering Reference
In-Depth Information
Using the geometry of the
M
/
EI
diagram, i.e. the semi-graphical approach, and
taking the moment of the area of the
M
/
EI
diagram between A and C about A we
have from Eq. (ii)
L
2
wL
2
8
EI
L
2
2
3
5
8
v
C
=−
which gives
5
wL
4
384
EI
v
C
=−
(see Eq. (v) of Ex. 13.4).
For the completely analytical approachwe express the bendingmoment
M
as a function
of
x
; thus
wx
2
2
wL
2
M
=
x
−
or
w
2
(
Lx
x
2
)
M
=
−
Substituting for
M
in Eq. (ii) we have
L
/
2
w
2
EI
(
Lx
2
x
3
)d
x
v
C
=−
−
0
which gives
Lx
3
3
−
L
/
2
x
4
4
w
2
EI
v
C
=−
0
Then
5
wL
4
384
EI
v
C
=−
E
XAMPLE
13.13
Figure 13.15(a) shows a cantilever beam of length
L
carrying a
concentrated load
W
at its free end. The section of the beam changes midway along
its length so that the second moment of area of its cross section is reduced by half.
Determine the deflection of the free end.
In this problem the bending moment and
M
/
EI
diagrams have different geometrical
shapes. Choosing the origin of axes at C, Eq. (13.10) becomes
x
A
d
v
d
x
x
C
d
v
d
x
A
M
EI
x
d
x
A
−
C
−
(
v
A
−
v
C
)
=
(i)
C
in which (d
v/
d
x
)
A
=
0,
x
C
=
0,
v
A
=
0. Hence
L
M
EI
x
d
x
v
C
=
(ii)
0
