Civil Engineering Reference
In-Depth Information
E
XAMPLE
15.5
Determine the rotation, i.e. the slope, of the beam ABC shown in
Fig. 15.11(a) at A.
W
A
B
C
EI
W
W
2
2
x
L
/2
L
/2
(a)
Unit moment
C
A
u
A
1
F
IGURE
15.11
Determination of the
rotation of a simply
supported beam at
a support using the
unit load method
1
R
v,C
R
v,A
L
L
L
(b)
The actual rotation of the beam at A produced by the actual concentrated load,
W
,is
θ
A
. Let us suppose that a virtual unit moment is applied at A before the actual rotation
takes place, as shown in Fig. 15.11(b). The virtual unit moment induces virtual support
reactions of
R
v,A
(
=
1/
L
) acting downwards and
R
v,C
(
=
1/
L
) acting upwards. The actual
internal bending moments are
W
2
x
M
A
=+
0
≤
x
≤
L
/
2
W
2
(
L
M
A
=+
−
x
)
L
/
2
≤
x
≤
L
The internal virtual bending moment is
1
L
x
M
v
=
1
−
0
≤
x
≤
L
The external virtual work done is 1
θ
A
(the virtual support reactions do no work as
there is no vertical displacement of the beam at the supports) and the internal virtual
work done is given by Eq. (15.21). Hence
L
/
2
L
d
x
L
2
x
1
L
d
x
x
)
1
1
EI
W
x
W
2
(
L
x
1
θ
A
=
−
+
−
−
(i)
0
L
/
2
