Force Couples and Couple Moment

A force couple consists of two parallel forces that are equal in magnitude, opposite in direction, and separated by a perpendicular distance. The resultant force is zero, but the couple produces a pure rotational effect.

Moment of a couple

The magnitude of the couple moment is

\[M=Fd,\]

where \(F\) is the magnitude of either force and \(d\) is the perpendicular distance between their lines of action, called the couple arm.

In a planar problem, the sign of \(M\) is assigned according to the sense of rotation. A common convention takes counterclockwise moments as positive and clockwise moments as negative.

Why the couple moment is independent of the reference point

Let the forces be \(\vec F\) and \(-\vec F\), applied at position vectors \(\vec r_1\) and \(\vec r_2\) measured from an arbitrary point \(O\). Their total moment about \(O\) is

\[\vec M_O=\vec r_1\times\vec F+\vec r_2\times(-\vec F)=(\vec r_1-\vec r_2)\times\vec F.\]

The result depends only on the relative position of the two lines of action, not on the location of \(O\). Therefore, the moment of a couple is the same about any point.

Vector representation

In three dimensions, the couple moment is represented by a vector \(\vec M\) perpendicular to the plane containing the two forces. Its direction is determined by the right-hand rule.

Because the couple moment is independent of the reference point, \(\vec M\) is a free vector: it may be translated parallel to itself without changing the external effect on a rigid body.

Equivalent couples

Two couples are mechanically equivalent if their moment vectors are equal:

\[\vec M_1=\vec M_2.\]

Thus, the force magnitude and arm may change while the product \(Fd\) and the moment direction remain the same.

Addition of couples

Several couples acting on a rigid body can be replaced by a single resultant couple whose moment is the vector sum

\[\vec M_R=\sum_i\vec M_i.\]

For coplanar couples, this becomes an algebraic sum of signed moments.

Example

Two opposite parallel forces of magnitude \(F=100\,\text{N}\) are separated by \(d=0.20\,\text{m}\). The couple moment magnitude is

\[M=100\cdot0.20=20\,\text{N}\cdot\text{m}.\]

A pair of \(50\,\text{N}\) forces separated by \(0.40\,\text{m}\) creates the same couple moment, provided the sense of rotation is unchanged.