Civil Engineering Reference
In-Depth Information
Alternatively, we could select the fixing moment,
M
A
(
M
2
), at A as the release. The
primary structure is then the simply supported beam shown in Fig. 16.16(a) where
R
A
=−
=
3
W
/2. The rotation at A may be found by any of the methods
previously described. They include the integration of the second-order differential
equation of bending (Eq. (13.3)), the moment-area method described in Section 13.3
and the unit load method (in this case it would be a unit moment). Thus, using the
unit load method and applying a unit moment at A as shown in Fig. 16.16(b) we have,
from the principle of virtual work (see Ex. 15.5)
W
/2 and
R
B
=
L
3
L
/
2
M
a
M
v
EI
M
a
M
v
EI
1
θ
A,0
=
d
x
+
d
x
(iii)
L
0
In Eq. (iii)
W
2
xM
v
=
1
L
x
M
a
=−
−
1(0
≤
x
≤
L
)
L
3
WL
2
3
L
2
M
a
=
Wx
−
M
v
=
≤
x
≤
0
Substituting in Eq. (iii) we have
L
W
2
EIL
x
2
)d
x
θ
A,0
=
(
Lx
−
0
from which
WL
2
12
EI
θ
A,0
=
(anticlockwise)
The flexibility coefficient,
θ
22
, i.e. the rotation at A (point 2), due to a unit moment at
A is obtained from Fig. 16.16(b). Thus
L
x
L
−
1
2
d
x
1
EI
θ
22
=
0
from which
L
3
EI
θ
22
=
(anticlockwise)
W
1
A
B
C
A(2)
B(1)
C
1/
L
1/
L
R
A
R
B
x
L
L
/2
F
IGURE
16.16
Alternative solution
for Ex. 16.6
(a)
(b)
