Civil Engineering Reference
In-Depth Information
Hence, substituting for
M
a
and
M
1
in Eq. (v), we have
L
w
2
EI
(
L
x
)
3
d
x
v
B,0
=
−
0
which gives
wL
4
8
EI
v
B,0
=
(vi)
We now apply a vertically downward unit load at the B end of the cantilever from
which the distributed load has been removed. The displacement,
v
B,1
, due to this unit
load is then, from Eq. (v)
L
1
EI
(
L
x
)
2
d
x
v
B,1
=
−
0
from which
L
3
3
EI
v
B,1
=
(vii)
The displacement due to
R
B
atBis
R
B
v
B,1
(
R
B
acts in the opposite direction to the
unit load) so that the total displacement,
v
B
, at B due to the uniformly distributed load
and
R
B
is, using the principle of superposition
−
v
B
=
v
B,0
−
R
B
v
B,1
=
0
(viii)
Substituting for
v
B,0
and
v
B,1
from Eqs (vi) and (vii) we have
wL
4
8
EI
−
R
B
L
3
3
EI
=
0
which gives
3
wL
8
as before. This approach is the flexibility method described in Section 16.1 and is, in
effect, identical to the method used in Ex. 13.18.
R
B
=
In Eq. (viii)
v
B,1
is the displacement at B in the direction of
R
B
due to a unit load
at B applied in the direction of
R
B
(either in the same or opposite directions). For a
beam that has a degree of statical indeterminacy greater than 1 there will be a series
of equations of the same form as Eq. (viii) but which will contain the displacements
at a specific point produced by the redundant forces. We shall therefore employ the
flexibility coefficient a
kj
(
k
1, 2,
...
,
r
) which we defined in Section 15.4
as the displacement at a point
k
in a given direction produced by a unit load at a point
j
in a second direction. Thus, in the above,
v
B,1
=
=
1, 2,
...
,
r
;
j
=
a
11
so that Eq. (viii) becomes
v
B,0
−
a
11
R
B
=
0
(ix)
