Coplanar force and couple

We have already seen in Section 2.4 that a force acting at one point of a rigid body is equivalent to the same force with a different (but parallel) line of action together with a couple. However, we shall study this in greater detail before considering several non-coplanar forces and couples.

Firstly, let us see with the aid of diagrams how a force and a coplanar couple may be replaced by a single force. Figure 8.1a shows a force and a couple both acting in the x, y plane. The force is F acting at P and the couple is made up from the two forces G and −G with their lines of action a distance d1 apart. A couple has the same effect wherever it acts in the plane. Hence, we can replace it by a couple at P, indicated in Figure 8.1b by a double-line vector, using the right-hand thread rule. The magnitude of the couple vector is ∣G∣d1. Since it acts in the positive z-direction, the vector moment of the couple is ∣G∣d1k, where k is the unit vector in the z-direction.

Finally, the force and couple at P is equivalent to just the force F at the point Q, still in the x, y plane, as shown in Figure 8.2. Q is positioned so that the moment of F about P is the same as the moment of the original couple.