Civil Engineering Reference
In-Depth Information
M
(a)
P
C
h
P
A
B
F
IGURE
3.3
Moments applied
to beams
M
Ph
(b)
BENDING MOMENT
In practice it is difficult to apply a pure bending moment such as that shown in
Fig. 3.3(a) to a beam. Generally, pure bending moments arise through the application
of other types of load to adjacent structural members. For example, in Fig. 3.3(b), a
vertical member BC is attached to the cantilever AB and carries a horizontal shear
load,
P
(as far as BC is concerned). AB is therefore subjected to a pure moment,
M
=
Ph
, at B together with an axial load,
P
.
TORSION
A similar situation arises in the application of a pure torque,
T
(Fig. 3.4(a)), to a beam.
A practical example of a torque applied to a cantilever beam is given in Fig. 3.4(b)
where the horizontal member BC supports a vertical shear load at C. The cantilever
AB is then subjected to a pure torque,
T
=
Wh
, plus a shear load,
W
.
All the loads illustrated in Figs 3.1-3.4 are applied to the various members by some
external agency and are therefore
externally applied loads
. Each of these loads induces
reactions in the support systems of the different beams; examples of the calculation of
support reactions are given in Section 2.5. Since structures are in equilibrium under
a force system of externally applied loads and support reactions, it follows that the
support reactions are themselves externally applied loads.
Now consider the cantilever beam of Fig. 3.2(a). If we were to physically cut through
the beam at some section 'mm' (Fig. 3.5(a)) the portion BC would no longer be able
to support the load,
W
. The portion AB of the beam therefore performs the same
function for the portion BC as does the wall for the complete beam. Thus at the
section mm the portion AB applies a force
W
and a moment
M
to the portion BC at
B, thereby maintaining its equilibrium (Fig. 3.5(b)); by the law of action and reaction
(Newton's Third Law of Motion), BC exerts an equal force systemon AB, but opposite
