Civil Engineering Reference
In-Depth Information
which from mathematical theory shows that the curve representing the variation in
bending moment is convex in the positive direction of bending moment. This may be
observed in the bending moment diagrams in Fig. 3.12(d), 3.15(d) and 3.16(e). In this
example the bending moment diagram for the complete beam is shown in Fig. 3.19(c)
and is again drawn on the tension side of the beam.
E
XAMPLE
3.11
A precast concrete beam of length
L
is to be lifted from the casting
bed and transported so that the maximum bending moment is as small as possible. If
the beam is lifted by two slings placed symmetrically, show that each sling should be
0
.
21
L
from the adjacent end.
The external load on the beam is comprised solely of its own weight, which is uni-
formly distributed along its length. The problem is therefore resolved into that of a
simply supported beam carrying a uniformly distributed load in which the supports
are positioned at some distance
a
from each end (Fig. 3.20(a)).
The shear force and bending moment diagrams may be constructed in terms of
a
using
the methods described above and would take the forms shown in Fig. 3.20(b) and (c).
Examination of the bending moment diagram shows that there are two possible posi-
tions for the maximum bending moment. First at B and C where the bending moment
is hogging and has equal values from symmetry; second at the mid-span point where
w
A
D
B
C
wL
wL
R
B
R
C
a
a
2
2
L
(a)
wL
wa
2
wa
Shear force
ve
ve
A
D
B
C
ve
ve
wa
wL
wa
2
(b)
F
IGURE
3.20
Determination of
optimum position
for supports in the
precast concrete
beam of Ex. 3.11
Bending moment
ve
A
B
C
D
ve
(c)
