Civil Engineering Reference
In-Depth Information
It is also convenient, since the flexibility coefficients are specified by numerical
subscripts, to redesignate
R
B
as
R
1
. Thus Eq. (ix) becomes
v
B,0
−
a
11
R
1
=
0
(x)
E
XAMPLE
16.6
Determine the support reaction at B in the propped cantilever
shown in Fig. 16.15(a).
1
W
W
A(2)
B(1)
A
B
C
A
B
C
M
A
(
M
2
)
EI
x
x
L
L
/2
L
L
/2
R
2
R
1
x
L
L
/2
F
IGURE
16.15
Propped cantilever
of Ex. 16.6
(a)
(b)
(c)
As in Ex. 16.5, the cantilever in Fig. 16.15(a) has a degree of statical indeterminacy
equal to 1. Again we shall choose the support reaction at B,
R
1
, as the indeterminacy;
the released or primary structure is shown in Fig. 16.15(b). Initially we require the
displacement,
v
B,0
, at B due to the applied load,
W
, at C. This may readily be found
using the unit load method. Thus from Eq. (iii) of Ex. 15.9
3
L
2
−
x
L
W
EI
v
B,0
=
−
{−
1(
L
−
x
)
}
d
x
0
which gives
7
WL
3
12
EI
v
B,0
=
(i)
Similarly, the displacement at B due to the unit load at B in the direction of
R
1
(Fig.
16.15(c)) is
L
3
3
EI
a
11
=
(use Eq. (vii) of Ex. 16.5)
Hence, since,
v
B,0
−
a
11
R
1
=
0
(ii)
we have
7
WL
3
12
EI
−
L
3
3
EI
R
1
=
0
from which
7
W
4
R
1
=
