Force Couples and Couple Moment

A force couple is a system of two parallel forces of equal magnitude, opposite direction, and different lines of action. The vector sum of the two forces is zero, but their moments produce a nonzero rotational effect.

Couple arm and couple moment

The couple arm $d$ is the shortest, perpendicular distance between the lines of action of the two forces.

The magnitude of the moment of a force couple equals the magnitude of either force $F$ multiplied by the couple arm $d$, the perpendicular distance between their lines of action:

$$M=Fd$$

In a planar problem, the sign is determined by the sense of rotation produced by the couple according to the adopted sign convention.

The SI unit of couple moment is N·m. A couple moment should not be confused with work or energy even though their SI dimensions have the same unit form.

Why the resultant force is zero

The two forces have equal magnitudes and opposite directions, so their vector sum is zero: $\vec F+(-\vec F)=0$. Their lines of action do not coincide, however, so the moments of the forces do not cancel. A couple therefore cannot be replaced by a single nonzero force.

Independence from the reference point

The sum of the moments of the two forces of a couple is the same about any reference point. The couple moment is therefore a free vector: in the rigid-body model, a couple may be moved to another location without changing its mechanical effect as long as the magnitude and direction of its moment remain unchanged.

Equivalent couples

Two couples are equivalent when their moment vectors are equal. Thus, a larger force acting with a smaller arm may be equivalent to a smaller force acting with a larger arm if the product $Fd$ remains the same.

Addition of couples

Several couples can be replaced by one couple whose moment is the vector sum of the original couple moments. In planar statics, this reduces to algebraic addition with the adopted moment signs.

Example

Two forces of 300 N form a couple with an arm $d=0.20$ m. The couple moment is $M=300\cdot0.20=60$ N·m. If the arm is increased to 0.30 m while the same moment is required, forces of only $F=60/0.30=200$ N are needed.

Key points

• the vector sum of the forces in a couple is zero;
• a couple produces a pure rotational effect;
• the couple moment equals $Fd$ and is independent of the reference point;
• couples with equal moment vectors are equivalent;
• in planar problems, couple moments are added algebraically.