Equivalent Couples

Consider a valve subjected to two couples in the same plane having the same moments.

If the magnitude, direction, and rotation sense of their moments are the same, the couples are equivalent.

First, denote the points of intersection of the lines of action of the two couples.

Then, move the first couple's two forces to the second couple's points of intersection and resolve them into components.

One pair of components has the same magnitude, line of action and opposite sense and can be canceled.

So, the first couple reduces to a new couple formed by forces S and -S.

Its moment equals the moment of force S about a point lying on the line of action of -S.

Using Varignon's theorem, the first couple's moment equals the sum of the moments of its components, which in turn equals the new couple's moment.

So, the new forces are equal to the second couple's forces; implying that the two couples are equivalent.

Similarly, two couples contained in parallel planes having the same moments are also equivalent.