Civil Engineering Reference
In-Depth Information
We choose the origin of the axes at the free end B of the cantilever. Equation (13.7)
then becomes
d
v
d
x
A
−
d
v
d
x
B
M
EI
d
x
B
=
A
or, since (d
v/
d
x
)
A
=
0
d
v
d
x
L
M
EI
d
x
−
B
=
(i)
0
Generally at this stage we decide which approach is most suitable; however, both
semi-graphical and analytical methods are illustrated here. Using the geometry of Fig.
13.13(b) we have
d
v
d
x
2
L
−
1
WL
EI
−
B
=
which gives
d
v
d
x
WL
2
2
EI
(compare with the value given by Eq. (iii) of Ex. 13.1. Note the change in sign due to
the different origin for
x
).
B
=
Alternatively, since the bending moment at any section
x
is
−
Wx
we have, from Eq. (i)
d
v
d
x
L
Wx
EI
d
x
−
B
=
−
0
which again gives
d
v
d
x
WL
2
2
EI
With the origin for
x
at B, Eq. (13.10) becomes
x
A
d
v
d
x
B
=
x
B
d
v
d
x
A
M
EI
x
d
x
A
−
B
−
(
v
A
−
v
B
)
=
(ii)
B
Since (d
v/
d
x
)
A
=
0 and
x
B
=
0 and
v
A
=
0, Eq. (ii) reduces to
L
M
EI
x
d
x
v
B
=
(iii)
0
Again we can now decide whether to proceed semi-graphically or analytically. Using
the former approach and taking the moment of the area of the
M
/
EI
diagram about
B, we have
2
L
−
2
3
L
1
WL
EI
v
B
=
which gives
WL
3
3
EI
v
B
=−
(compare with Eq. (v) of Ex. 13.1)
