Couple
Hiroki Okubo
M
O
−F
O
F
a
rB
B
r
rA
A
d
M
M
M
−2F
−F
−F
d/2
F
d
F
−F
d
F
2F
−F
d
F
(a)
Figure 2: Equivalent couple
(b)
2
M
We may express the moment of a couple by using
vector algebra. With the cross-product notation,
the combined moment about point O of the forces
forming the couple is M = r A × F + r B × (−F ) =
(r A −r B )×F where r A and r B are position vectors
which run from point O to arbitrary points A and
B on the lines of action of F and −F , respectively,
we can express M as M = r × F .
O
(c)
Figure 1: Couple
1
Vector algebra method
3
Equivalent couples
Changing the values of F and d does not change
a given couple as long as the product F d remains
the same. A couple is not affected if the forces act
in a different but parallel plane. Figure 2 shows
four different configurations of the same couple M .
The couples are equivalent and are described as the
same vector.
Introduction
The moment produced by two equal, opposite, and
noncollinear forces is called a couple. Figure 1(a)
shows the action of two equal and opposite forces F
and −F with a distance d apart. These two forces
cannot be combined into a single force because their
sum in every direction is zero. Their only effect is
to produce a tendency of rotation. The combined
moment of the two forces about an axis normal to
their plane and passing through any point such as
O in their plane is the couple M . This couple has a
magnitude M = F (a + d) − F a = F d. Its direction
is counterclockwise when viewed from above.
4
Force-couple systems
The effect of a force acting on a body is the tendency to push or pull the body in the direction of
the force, and to rotate the body about any fixed
axis which does not intersect the line of the force.
We can represent this dual effect more easily by replacing the given force by an equal parallel force
and a couple to compensate for the change in the
moment of the force. The replacement of a force by
1
B
B
B
F
ment M are replaced by an equal force F at the
point of yA = −0.4 m.
F
−F d
F
F
A
M=Fd
A
A
References
[1] J. L. Meriam and L. G. Kraige, (2011), Engineering Mechanics, Volume 1, Statics, 7th edition, Wiley
Figure 3: Force-couple systems
y
200 N
x
O
80 Nm
Figure 4: Example of force-couple system
a force and a couple is illustrated in Fig. 3, where
the given force F acting at point A is replaced by
an equal force F at some point B and the counterclockwise couple M = F d. The transfer is seen
in the middle figure, where the equal and opposite
forces F and −F are added at point B without introducing any net external effects on the body. We
see that the original force at A and the equal and
opposite one at B constitute the couple M = F d.
Thus, we have replaced the original force at A by
the same force acting at a different point B and a
couple, without altering the external effects of the
original force on the body. The combination of the
force and couple in the right-hand part of Fig. 3 is
referred to as a force-couple system.
5
Example
Replace the force-couple system at point O by a
single force. Specify the coordinate yA of the point
on the y-axis through which the line of action of
this resultant force passes.
Solution. The couple is M = −80k N · m. When
the forces F = −200i N and −F produce the clockwise couple the moment arm can be calculated by
80/200 = 0.4 m. Thus, the given force F and mo2