Imagine this fire hydrant, fixed to the ground. 
and there’s a force F acting on it as shown. From experience, we can tell that if we replace this force by 
these two forces, each with half the magnitude, placed symmetrically about the central axis of the hydrant, 
these two forces will create the same effect as the original F force.
Even if we replace the forces by these three, again they create the same effect. 
By same effect I mean that the forces create the same tendency to push the fire hydrant down, and also, the ground will generate the 
same force to support the fire hydrant, preventing it from going down.
These several force systems are known to be equivalent systems. 
Now let’s imagine the force F acts on the fire hydrant this way. 
Now the force creates a translational tendency to push the fire hydrant to the right, 
and also it creates a clockwise rotational tendency for the fire hydrant to fall to the right.
For this fire hydrant to stay still, as a response, the ground must create a force supporting the 
fire hydrant, pointing to the left, and also a moment to cancel out the rotational effect.
We can add a pair of cancelling forces to this fire hydrant without changing the load status. 
But notice,
these two forces now create a couple moment, and can be represented as: 
Note the cross symbol indicates that the direction of the moment vector is pointing into the screen. You can imagine an arrow 
shooting into the screen. Now this force still provides a translational tendency to push the fire hydrant to the right, 
while this couple moment creates the clockwise rotational effect on the fire hydrant as well. 
So again, in order to keep the fire hydrant static, the ground must create a force pointing to the left, 
and a counterclockwise moment to cancel out the rotational effect.
So this force moment system
is the equivalent system as the previous single force system.
A system is equivalent
if the external effects it produces on a body are the same as those caused by the original force and couple moment system. 
In the class of statics, since members are not moving, 
we say the systems are equivalent
if they induce the same support reactions at the supports.
A load with multiple forces and couple moments acting on multiple locations can be replaced by
a single force and a single couple moment acting on a single point. 
We want to do so to help calculate the support reactions.
Let’s imagine the fire hydrant is subjected to multiple forces and multiple couple moments acting on multiple points.
We want to replace all of these by a single force and a single couple moment placed at a certain point, 
say point O.
The single force is simply the resultant force of all these three original forces, and it’s easily calculated through vector addition. 
For the resultant moment, we need to first calculate the individual moment caused by each force about point O, 
add them together,
and then add all of the free couple moments together, in this case, only two, M_1 and M_2.
And then we add the total moment caused by the forces and the total couple moments together, 
and this is the total moment at point O. 
And therefore we replaced the original multi-force multi-moment load system by a single force single moment system,
acting on a single point, point O.
You can do this for any arbitrary point. Note that the resultant force will always be the same, 
but the resultant moment will depend on what reference point you choose. 
In some special situation, the resultant force vector and the resultant moment vector are perpendicular to each other, as shown here. 
You can further reduce the moment by placing the force away from point O, 
say at a distance d. 
d equals to the magnitude of the resultant moment M_RO divided by the resultant force F_R.
The reason is because, this way, force F_R is creating a moment about point O that equals to F_R times d, 
which is M_RO.
Therefore this way the system is reduced to a single force system.
Let's look at this example. This member is  subjected to multiple force and moment, and we need to replace that by a single force that 
is equivalent to the original applied force and moment system.
To do that let's first replace the applied force and moment system with an equivalent system that has one force and one couple moment.
The force is the resultant force which is the summation of all the applied force vectors.
And the resultant moment can be summarized about any point, in this case we choose point A. This resultant moment has two parts: 
the first part is the moment caused by all the forces about point A; the second part
includes all the applied free couple moments.
Because this is a 2D problem it is easier to apply scalar formulations instead. To do that 
I'm going to first resolve all the applied forces into their vertical and horizontal components.
And now I can summarize the resultant force along the x direction, along the y direction as well as the resultant moment about point A.
And this equivalent force-moment system must be placed at point A because I summarized the moment at point A. 
Notice that the two force vector components can be easily replaced by their resultant force vector.
However since the moment vector is perpendicular to the force vectors, again this is always true for 2D problems, we can move 
the force vectors to a different location so that they can create the same moment about point A.
There are other ways to move the forces as well, but according to the problem statement we need to move the forces along the AB segment.
Now we need to determine the location x.
Since only the vertical force component can create moment about point A, and we know that it must create a -0.76 kilo-newton meter 
moment about point A, based on that information we can calculate x to be
the magnitude of the moment divided by the magnitude of the vertical force component, 
to be 2.3 meter. And we can also calculate the magnitude of the resultant force.
And now we have successfully
replaced the origional applied force moment system with a single force.
Its location is specified.
And this is the equivalent system as the original system.
