Principles of Statics - Structural and Stress Analysis - page 27

I 1

E

F 1

D

F 2

F 2

F 1

B

A

R 12

R 123
( R )

C

F IGURE 2.10
Resultant of a
system of
non-concurrent
forces

F

F 3

F 3

I 2

(a) (b)

Again, the law of the polygon of forces may be used in the analysis of plane, pin-jointed
trusses where several members meet at a joint but where no more than two forces are
unknown in magnitude.

THE RESULTANT OF A SYSTEM OF NON-CONCURRENT
FORCES

In most structural problems the lines of action of the different forces acting on the
structure do not meet at a single point; such a force system is non-concurrent.

Consider the system of non-concurrent forces shown in Fig. 2.10(a); their resultant
may be found graphically using the parallelogram of forces as demonstrated in Fig.
2.10(b). Produce the lines of action of F 1 and F 2 to their point of intersection, I 1 .
Measure I 1 A F 2 to the same scale, then complete the parallelogram

I 1 ACB; the diagonal CI 1 represents the resultant, R 12 ,of F 1 and F 2 . Now produce
the line of action of R 12 backwards to intersect the line of action of F 3 at I 2 . Measure
I 2 D

= F 1 and I 1 B =

F 3 to the same scale as before, then complete the parallelogram
I 2 DEF; the diagonal I 2 E

= R 12 and I 2 F =

R ,
the resultant of F 1 , F 2 and F 3 . Note that only the line of action and the magnitude of
R can be found, not its point of action, since the vectors F 1 , F 2 and F 3 in Fig. 2.10(a)
define the lines of action of the forces, not their points of action.

= R 123 , the resultant of R 12 and F 3 . It follows that R 123 =

If the points of action of the forces are known, defined, say, by coordinates referred
to a convenient xy axis system, the magnitude, direction and point of action of their
resultant may be found by resolving each force into components parallel to the x and
y axes and then finding the magnitude and position of the resultants R x and R y of each
set of components using the method described in Section 2.3 for a system of parallel
forces. The resultant R of the force system is then given by

R x +

R y R =