Civil Engineering Reference
In-Depth Information
w
A
B
EI
x
L
(a)
1 (Unit load)
A
B
F
IGURE
15.10
Deflection of the free
end of a cantilever
beam using the unit
load method
B
(b)
the unit load is, from Fig. 15.10(b), 1
υ
B
. The deflection,
υ
B
, is assumed to be caused
by bending only, i.e. we are ignoring any deflections due to shear. The internal virtual
work is given by Eq. (15.21) which, since only one member is involved, becomes
L
M
A
M
v
EI
W
i,
M
=
d
x
(i)
0
The virtual moments,
M
v
, are produced by a unit load so that we shall replace
M
v
by
M
1
. Then
L
M
A
M
1
EI
W
i,
M
=
d
x
(ii)
0
At any section of the beam a distance
x
from the built-in end
w
2
(
L
x
)
2
M
A
=−
−
M
1
=−
1(
L
−
x
)
Substituting for
M
A
and
M
1
in Eq. (ii) and equating the external virtual work done by
the unit load to the internal virtual work we have
L
w
2
EI
(
L
x
)
3
d
x
1
υ
B
=
−
0
which gives
1
4
(
L
x
)
4
L
0
w
2
EI
υ
B
=−
−
so that
wL
4
8
EI
υ
B
=
(as in Ex. 13.2)
Note that
υ
B
is in fact negative but the positive sign here indicates that it is in the same
direction as the unit load.
