Civil Engineering Reference
In-Depth Information
Note that vectors obey the
commutative law
, i.e.
F
2
+
F
1
=
F
1
+
F
2
The actual magnitude and direction of
R
may be found graphically by drawing the
vectors representing
F
1
and
F
2
to the
same scale
(i.e. OB and BC) and then completing
the triangle OBC by drawing in the vector, along OC, representing
R
. Alternatively,
R
and
θ
may be calculated using the trigonometry of triangles, i.e.
R
2
F
1
+
F
2
+
=
2
F
1
F
2
cos
α
(2.1)
and
F
1
sin
α
F
2
+
tan
θ
=
(2.2)
F
1
cos
α
In Fig. 2.3(a) both
F
1
and
F
2
are 'pulling away' from the particle at O. In Fig. 2.4(a)
F
1
is a 'thrust' whereas
F
2
remains a 'pull'. To use the parallelogram of forces the system
must be reduced to either two 'pulls' as shown in Fig. 2.4(b) or two 'thrusts' as shown
in Fig. 2.4(c). In all three systems we see that the effect on the particle at O is the same.
As we have seen, the combined effect of the two forces
F
1
and
F
2
acting simultaneously
is the same as if they had been replaced by the single force
R
. Conversely, if
R
were to
be replaced by
F
1
and
F
2
the effect would again be the same.
F
1
and
F
2
may therefore
be regarded as the
components
of
R
in the directions OA and OB;
R
is then said to
have been
resolved
into two components,
F
1
and
F
2
.
Of particular interest in structural analysis is the resolution of a force into two com-
ponents at right angles to each other. In this case the parallelogram of Fig. 2.3(b)
F
2
(pull)
O
R
F
1
F
1
(thrust)
O
F
2
(a)
(b)
F
2
O
R
F
1
F
IGURE
2.4
Reduction of a
force system
(c)
