Civil Engineering Reference
In-Depth Information
C
F
1
R
E
(
R
)
R
F
1
O
O
B
F
2
F
2
R
E
(
R
)
Equilibrant of
F
1
and
F
2
F
IGURE
2.8
Equilibrant of two
concurrent forces
(a)
(b)
O, as shown in Fig. 2.8(a).
R
E
is known at the
equilibrant
of
F
1
and
F
2
and the particle
at O will then be in
equilibrium
and remain stationary. In other words the forces
F
1
,
F
2
and
R
E
are in equilibrium and, by reference to Fig. 2.3(b), we see that these three
forces may be represented by the triangle of vectors OBC as shown in Fig. 2.8(b). This
result leads directly to the law of the
triangle of forces
which states that:
If three forces acting at a point are in equilibrium they may be represented in magnitude
and direction by the sides of a triangle taken in order.
The law of the triangle of forces may be used in the analysis of a plane, pin-jointed
truss in which, say, one of three concurrent forces is known in magnitude and direction
but only the lines of action of the other two. The law enables us to find the magnitudes
of the other two forces and also the direction of their lines of action.
The above arguments may be extended to a system comprising any number of concur-
rent forces. In the force systemof Fig. 2.6(a),
R
E
, shown in Fig. 2.9(a), is the equilibrant
of the forces
F
1
,
F
2
,
F
3
and
F
4
. Then
F
1
,
F
2
,
F
3
,
F
4
and
R
E
may be represented by the
force polygon OBCDE as shown in Fig. 2.9(b).
F
2
D
R
F
3
F
4
F
3
C
O
F
1
E
F
2
R
E
F
IGURE
2.9
Equilibrant of a
number of
concurrent forces
F
4
R
E
(
R
)
O
F
1
B
(a)
(b)
The law of the
polygon of forces
follows:
If a number of forces acting at a point are in equilibrium they may be represented in
magnitude and direction by the sides of a closed polygon taken in order.
