If the load on a column is applied through the center of gravity of its cross section, it is called an axial load.
Axial force is the compression or tension force acting in a member. If the axial force acts through the centroid of the member it is called concentric loading. If the force is not acting through the centroid it's called eccentric loading. Eccentric loading produces a moment in the beam as a result of the load being a distance away from the centroid.
An interesting example of axial loading can be seen in the image to the right. Imagine a pogo stick person who is trying to jump with only one foot, what would happen? Well they wouldn't stay upright. The ideal way to jump on a pogo stick is to ensure that the force is acting concentric to the pogo sticks centroid. If you are off by even a bit the moment will cause the pogo stick to react and you won't jump straight.
This will be easiest explained through an example.
Say you have a 12"x12"x10'-0" wood column supporting a platform. For the sake of this argument we'll assume the platform weighs 325 lbs (if i don't say this I'll get emails saying this example is wrong (trust me it happens)). On top of the platform is a 90- 9 lb bowling balls. What is the axial force acting at the top of the 12 x 12? What is the Axial Force at the bottom of the 12" x 12"?
Since the platform is weightless the only load acting on the column is the 90 bowling balls:
A = 90 * 9 lbs + 325 lbs = 1135 lbs
Is the answer still 1135 lbs? No, because the weight of the beam now needs to be considered! The new axial force can be found below:
A = 90 * 9 lbs + 325 lbs + (11.5" * 11.5" / 144 * 10'-0") * 35 pcf = 1456 lbs
Note: A wooden 12x12 actually has a Sawn Lumber Dimension of 11.5"
Told you it would be easier with an example. As you can see, the Axial Force is usually one of the easier forces to calculate. It is most often necessary to know for either columns or load combinations.
If the athlete on the pogo stick above stopped pogo-ing, what would be the axial force acting on the pogo stick?