Civil Engineering Reference
In-Depth Information
examples, its net area is zero since there is no change of slope between the ends of the
beam and
EI
is constant (see Eq. (13.7)). This condition is satisfied by
M
A
=
M
B
.
Let us now assume an origin for
x
at A; Eq. (13.10) becomes
x
B
d
v
d
x
x
A
d
v
d
x
A
−
B
M
EI
x
d
x
B
−
(
v
B
−
v
A
)
=
(i)
A
in which (d
v
/d
x
)
A
=
(d
v
/d
x
)
B
=
0,
v
A
=
0 and
v
B
=−δ
. Hence Eq. (i) reduces to
L
M
EI
x
d
x
δ =
0
Using the semi-graphical approach and taking moments of areas about A we have
1
2
L
2
M
A
EI
L
6
+
1
2
L
2
M
A
EI
5
6
L
δ =−
which gives
6
EI
δ
M
A
=
(hogging)
L
2
It follows that
6
EI
δ
M
B
=
(sagging)
L
2
The effect of building in the ends of a beam is to increase both its strength and its
stiffness. For example, the maximum bending moment in a simply supported beam
carrying a central concentrated load
W
is
WL
/4 but it is
WL
/8 if the ends are built-in. A
comparison of the maximum deflections shows a respective reduction from
WL
3
/
48
EI
to
WL
3
/
192
EI
. It would therefore appear desirable for all beams to have their ends
built-in if possible. However, in practice this is rarely done since, as we have seen,
settlement of one of the supports induces additional bending moments in a beam. It
is also clear that such moments can be induced during erection unless the supports
are perfectly aligned. Furthermore, temperature changes can induce large stresses
while live loads, which produce vibrations and fluctuating bending moments, can have
adverse effects on the fixity of the supports.
One method of eliminating these difficulties is to employ a double cantilever construc-
tion. We have seen that points of contraflexure (i.e. zero bending moment) occur at
sections along a fixed beam. Thus if hinges were positioned at these points the bend-
ing moment diagram and deflection curve would be unchanged but settlement of a
support or temperature changes would have little or no effect on the beam.
PROBLEMS
P.13.1
The beam shown in Fig. P.13.1 is simply supported symmetrically at two points
2m from each end and carries a uniformly distributed load of 5 kN/m together with
