Civil Engineering Reference
In-Depth Information
AX
1
in Fig. 3.17(b) or X
2
C in Fig. 3.17(c) we see that
M
0
L
S
AB
=
S
BC
=−
(i)
The shear force is therefore constant along the length of the beam as shown in
Fig. 3.17(d).
Now considering the moment equilibrium about X
1
of the length AX
1
of the beam in
Fig. 3.17(b)
M
0
L
x
M
AB
−
=
0
or
M
0
L
x
M
AB
=
(ii)
M
AB
therefore increases linearly from zero at A (
x
=
0) to
+
3
M
0
/
4atB(
x
=
3
L
/
4).
From Fig. 3.17(c) and taking moments about X
2
we have
M
0
L
(
L
M
BC
+
−
x
)
=
0
or
M
0
L
(
x
M
BC
=
−
L
)
(iii)
M
BC
therefore decreases linearly from
L
); the
complete distribution of bending moment is shown in Fig. 3.17(e). The deflected form
of the beam is shown in Fig. 3.17(f) where a point of contraflexure occurs at B, the
section at which the bending moment changes sign.
−
M
0
/4 at B (
x
=
3
L
/4) to zero at C (
x
=
In this example, as in Ex. 3.8, the exact form of the deflected shape cannot be deduced
from the bending moment diagram without analysis. However, using the method of
singularities described in Section 13.2, it may be shown that the deflection at B is
negative and that the slope of the beam at C is positive, giving the displaced shape
shown in Fig. 3.17(f).
3.5 L
OAD
,S
HEAR
F
ORCE AND
B
ENDING
M
OMENT
R
ELATIONSHIPS
It is clear from Exs 3.4-3.9 that load, shear force and bending moment are related.
Thus, for example, uniformly distributed loads produce linearly varying shear forces
and maximum values of bending moment coincide with zero shear force. We shall now
examine these relationships mathematically.
The length of beamshown inFig. 3.18(a) carries a general systemof loading comprising
concentrated loads and a distributed load
w
(
x
). An elemental length
x
of the beam
is subjected to the force and moment system shown in Fig. 3.18(b); since
δ
δ
x
is very
