When two couples produce a moment with the same magnitude and direction, they are equivalent couples.
Resultant Moment of Couple Force
Since the moment of couple are vectors their resultant can be determined by vector addition.
Since moment of couple are free vectors, we can join their tails and find resultant.
MR = M1 + M2
Simplification of a force and couple system
Sometimes it is convenient to reduce a system of forces and couple moments acting on a body by replacing it with an equivalent system, consisting of a single resultant force acting on a specified point and a resultant couple moment.
Equivalent Systems: If the external effects it produces on a body are the same as those caused by original force and couple moment system.
Note: If F is moved along the LOA, the reaction force will remain the same.
Principle of Transmissibility:
Force acting on a body is a sliding vector since it can be applied at any point along its LOA.
MOVING A FORCE ON ITS LOA
When a force is moved along its LOA, there is no external effects.
MOVING A FORCE OFF ITS LOA
A force can be moved from A to B provided a couple moment M is added to maintain an equivalent system.
Finding the Resultant of a Force and Couple System
When several forces and moment of couple forces acting on a body, you can move each force and its associated couple moments to a common point O. Now you can add all these forces and couple moments together and find one resultant force-couple moment pair.
If the force system lies in the x-y plane (2D) and any moment of couple force perpendicular to the plane then the above equations reduce to the following:
If F and M are perpendicular to each other, then the system can be further reduced to single force F by simply moving O to P.
For a better understand of the topic you visit the first part of this topic.