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Section 4.5 Couples

The moments we have considered so far were all caused single forces producing rotation about a moment center. In this section we will consider another type of moment, called a couple.

A couple consists of two parallel forces, equal in magnitude, opposite in direction, and non-coincident. Couples are special because the pair of forces always cancel each other, which means that a couple produces a rotational effect but never translation. For this reason, couples are sometimes referred to as “pure moments.” The strength of the rotational effect is called the moment of the couple or the couple-moment.

When a single force causes a moment about a point, the magnitude depends on the magnitude of the force and the location of the point. In contrast, the moment of a couple is the same at every point and only depends on magnitude of the opposite forces and the distance between them.

For example, consider the interactive where two equal and opposite forces with different lines of action form a couple. The moment of this couple is found by summing the moments of the two forces about arbitrary moment center \(A,\) applying positive or negative signs for each term according to the right hand rule. The moment of the couple is always

\begin{align} M \amp= Fd_\perp \tag{4.5.1} \end{align}

where \(d_\perp\) is the perpendicular distance between the lines of action of the forces.

Figure 4.5.1. Moment of a couple.

In two dimensions, couples are represented by a curved arrow indicating the direction of the rotational effect. Following the right-hand rule the value will be positive if the moment is counter-clockwise and negative if it is clockwise. In three dimensions, a couple is represented by a normal vector arrow.

When adding moments to find the total or resultant moment, you must include couple-moments as well the \(\vec{r}\times\vec{F}\) moments. In equation form, we could express this as:

\begin{equation*} \Sigma M_P = \Sigma (\vec{r}\times\vec{F} ) + \Sigma (\vec{M}_\text{couple}) \end{equation*}

In this section we have shown that couples produce the same moment at every point on the body. This means that the external effect of couples is location independent. Because the moment of a couple is location independent, the moment vector is not bound to any particular point and for this reason is a free vector.

We will learn in Chapter 8 that moving a couple around on a rigid body does affect the internal loads or stresses inside a body, but changing the location of a couple does not change the external loading or reactions.