what is a couple in statics
When you turn a steering wheel, one hand pushes up while the other pulls down. The two hands apply forces of equal size and opposite direction, so they do not shove the wheel either way and they only turn it. Two forces like this, equal in size, opposite in direction, and acting along parallel lines a distance apart, are called a couple. Adding the two forces gives 0, so they do not translate the body, yet a moment that tends to rotate it clearly remains. What is more, this moment is the same no matter where you place the axis. So the moment of a couple is a free quantity, not tied to a location. Start by growing the two forces and check that the resultant is 0 while the rotation stays alive.
Two forces of equal size and opposite direction sit a distance apart along parallel lines. The upper force points right and the lower one points left, and when you grow them they always grow together. Because they point opposite ways, adding them as vectors gives exactly 0 every time. The resultant is 0, so they drag the body in no direction at all, and yet, like the curved arrow in the middle, the effect that tends to rotate the body is clearly alive. This is the heart of a couple. The sum of the forces is 0 while the moment is not 0, a combination that produces a pure turning effect, rotation with no translation.
Why is the rotation pure. Comparing with a single force makes it plain, and switching the two with the buttons shows the difference. A single force acting off the center of the body both shoves the body along and spins it, so translation and rotation are mixed. A couple has a resultant of 0, so it cannot move the center at all. What remains is rotation only, a pure spin that turns in place. The turn a figure skater makes with both arms, and the force the fingers give when a screwdriver turns a screw, are close to couples. A couple is what you use when you want to turn something without moving it.
A couple has a special property that an ordinary moment lacks. The size of its moment is the same no matter where you place the axis, whereas the moment of a single force changes with the point you measure about, and a couple does not. Drag the reference point around. Whether the point sits between the two forces, outside them, or far away, the moment of the couple is always F times the distance between the two forces. One force moves farther from the reference and its moment grows, while the other moves closer and its moment shrinks, and the two changes cancel exactly. So the moment of a couple is a free vector, not tied to a point of application, simply a turning effect with a size and a sense.
The size of a couple is simple. Multiply the size of one force by the perpendicular distance between the two forces and that is M = F d, and the point is that the distance is the spacing of the pair, not the distance of one force from some chosen point. Change the force and the spacing with the two sliders and the couple grows the same way whether you raise the force or spread the two forces farther apart. So even a small force can make a large turning effect if it is spread far enough, and gripping both handles of a large wrench and turning them opposite ways is easier than pushing with one hand. The unit is N·m, the same as a moment, and the sign follows the same agreement, counterclockwise positive and clockwise negative.
Finally there is a very useful trick. When you want to move a force, parallel to itself, to a different spot off its line of action, just moving it would change the turning effect, so you attach a couple that compensates for the move. Then the effect stays exactly the same as the original force, and this is called a force-couple system. Drag the distance you move the force and the force at the new spot is unchanged, while a couple of size M = F e comes along with it. A complicated load can be rewritten as one force at a point plus one couple, which is why rigid-body analysis uses this so often.