Civil Engineering Reference
In-Depth Information
the point. Thus, in Fig. 2.12, the moment,
M
,of
F
about O is given by
M
=
Fa
(2.5)
where '
a
' is known as the
lever arm
or
moment arm
of
F
about O; note that the units
of a moment are the units of force
×
distance.
It can be seen from the above that a moment possesses both magnitude and a rota-
tional sense. For example, in Fig. 2.12,
F
exerts a clockwise moment about O. A
moment is therefore a vector (an alternative argument is that the product of a vec-
tor,
F
, and a scalar,
a
, is a vector). It is conventional to represent a moment vector
graphically by a double-headed arrow, where the direction of the arrow designates a
clockwise moment when looking in the direction of the arrow. Therefore, in Fig. 2.12,
the moment
M
(
Fa
) would be represented by a double-headed arrow through Owith
its direction into the plane of the paper.
=
Moments, being vectors, may be resolved into components in the same way as forces.
Consider the moment,
M
(Fig. 2.13(a)), in a plane inclined at an angle
θ
to the
xz
plane. The component of
M
in the
xz
plane,
M
xz
, may be imagined to be produced
by rotating the plane containing
M
through the angle
θ
into the
xz
plane. Similarly,
the component of
M
in the
yz
plane,
M
yz
, is obtained by rotating the plane containing
M
through the angle 90
−
θ
. Vectorially, the situation is that shown in Fig. 2.13(b),
where the directions of the arrows represent clockwise moments when viewed in the
directions of the arrows. Then
M
xz
=
M
cos
θ
M
yz
=
M
sin
θ
The action of a moment on a structural member depends upon the plane in which it
acts. For example, in Fig. 2.14(a), the moment,
M
, which is applied in the longitu-
dinal vertical plane of symmetry, will cause the beam to bend in a vertical plane. In
Fig. 2.14(b) the moment,
M
, is applied in the plane of the cross section of the beam
and will therefore produce twisting; in this case
M
is called a
torque
.
y
M
yz
M
xz
M
cos
u
M
M
M
xz
x
u
u
F
IGURE
2.13
Resolution of a
moment
z
M
yz
M
sin
u
(a)
(b)
