Civil Engineering Reference
In-Depth Information
x
F
1
F
2
F
1
R
F
2
R
a
F
IGURE
2.17
Resultant of a
system of parallel forces
(a)
(b)
The position of the line of action of
R
may be found using the principle stated above,
i.e. the sum of the moments of
F
1
and
F
2
about any point must be equivalent to the
moment of
R
about the same point. Thus fromFig. 2.17(a) and taking moments about,
say, the line of action of
F
1
we have
F
2
a
=
Rx
=
(
F
1
+
F
2
)
x
Hence
F
2
F
1
+
x
=
a
(2.7)
F
2
Note that the action of
R
is
equivalent
to that of
F
1
and
F
2
, so that, in this case, we
equate clockwise to clockwise moments.
The principle of equivalence may be extended to any number of parallel forces irre-
spective of their directions and is of particular use in the calculation of the position of
centroids of area, as we shall see in Section 9.6.
E
XAMPLE
2.1
Find the magnitude and position of the line of action of the resultant
of the force system shown in Fig. 2.18.
In this case the polygon of forces (Fig. 2.6(b)) degenerates into a straight line and
R
=
2
−
3
+
6
+
1
=
6kN
(i)
Suppose that the line of action of
R
is at a distance
x
from the 2 kN force, then, taking
moments about the 2 kN force
Rx
=−
×
+
×
+
×
3
0
.
6
6
0
.
9
1
1
.
2
Substituting for
R
from Eq. (i) we have
6
x
=−
1
.
8
+
5
.
4
+
1
.
2
