Thanks to Msquared (I think it was his article)
Here is a post a made a while ago.
OK, I am going to get a bit long winded here. I understand you want a short simple answer but I believe to fully understand the answer you have to look at the whole picture. This topic comes up often and I want everyone to understand it. I am sure some sort of can of worms will be opened over this but I want racers to learn and think about what they are doing and why they are doing it.
In a nut shell a softer axle will slow weight transfer and allow for less chassis flex keeping the RF planted more. This can in some instances free up the kart. A stiffer axle will increase weight transfer and plant the RR more.
I’m definitely not an expert on the practical application of different axle stiffness, but I suspect the ‘correct’ axle stiffness will very much depend on available grip and the basic rigidity of the rest of the chassis. I tend to think that ideally you want the axle to be as stiff as possible for the application and the chassis setup. However, if the chassis is too rigid (mechanically and/or geometrically) for the forces involved (basically the force created by available grip and driver weight) and the LR is tending to reload prematurely because of this excessive chassis rigidity (creating a high front roll stiffness) then short of changing to a different chassis it might be advantageous to use a softer axle. As such, I suspect needing to use a softer axle is to some degree ‘making up’ for a chassis that isn’t quite right for the grip conditions and driver weight.
I feel that this is an area of chassis tuning that is very overlooked by most and could be a very useful tuning tool not only for class but also for driver size.
My best take on this subject is that a stiffer spring will transfer more weight and a softer spring will transfer less weight. This is accepted roll couple theory. So, softening the load path (spring) from the CG to the RR by using a softer axle will reduce the DIFFERENCE in the front / rear load path stiffness (and thus the difference in roll stiffness). This should take some percentage of the transient weight transfer away from the RR and redistribute it to the RF. The CG/RF load path will now be relatively more rigid compared to the softened CG/RR load path, so somewhat more weight will transfer to the RF. However the CG/RF load path will be no more rigid in ABSOLUTE terms so the additional weight transfer may be enough to increase flex at the RF corner of the chassis or the waist and improve LR unloading. Improved LR rear unloading should lessen understeer and create a higher acceleration (G force) which will further assist in twisting the chassis and keeping the LR unloaded. To be perfectly clear, I’m talking here about LR unloading due to acceleration and weight transfer, not from the jacking effect. To be even more perfectly clear, I’m speculating on the mechanics of this to some degree.
The basic idea in chassis tuning as far as the rear end is concerned, is to get the chassis axle combo to put force on the RR at the correct rate and achieve the optimum amount of axle flex to hook up just right for a given situation. More flex is achieved through the use of shorter hubs and/or a thinner axle, and less flex achieved with a stiffer axle/more hub length. For a given frame stiffness, you will usually want a softer axle hub combo for high grip situations, and stiffer for low grip, however it is also possible to use a very stiff axle and tune the chassis for the correct amount of flex, as the overall flex is a combination of the two.
Axles have a sweet spot where they flex enough and give a free rear end but not so much that they create binding. The best way to know if your axle is too stiff/soft is to change your hub length. The same holds true for softer, that is, shorter hub makes it softer and if it feels better with the shorter hub, you are probably too stiff on the axle. I have put on a very long hub at the RR after a rain storm and simply killed the competition.
I think the answer to this question has to do with a few factors and so it can't be a generic answer. If you think about a softer or stiffer axle as it relates to spring rates and their effect on weight transfer it will help with finding a solution.
Softer springs tend to slow the weight transfer down and not transfer as much weight to the tire. This sometimes produces less grip, most of the times I've seen this in slower classes. On the other hand the slow weight transfer sometimes produces more grip when there's not much grip in the tire or the surface. It's one of those things that depends on a lot of other variables. You are changing roll stiffness and roll couple distribution.
With a stiff spring rate weight transfers faster and more of it transfers. This can also either produce more grip or lose you more grip depending on the variables involved. A tire produces more grip the more weight you put on it but at some point the amount of weight overcomes the amount of available grip. This is why sometimes if you stiffen the kart up a lot in high grip conditions it frees up. You have put enough weight into the tire that it breaks free. Highly unlikely on high grip dirt tracks.
I think a lot of people overlook what the axle flex does to the chassis. You have to remember, the axle is connected to the frame via the bearing hangers, and so by changing the axle, you are also changing the amount of flex that the frame will have at the rear end based on axle stiffness. This is why I believe some karts like soft axles to free the kart up because a stiffer kart needs a softer axle to allow enough frame flex which will free up the kart. Conversely, a softer kart would need a stiffer axle to allow the rear end to stay stiffer instead of folding up (too much flex) and therefore causing excessive side bite, making the kart tight. I believe this is also why on some karts changing the hubs can do the opposite, that is, putting on a shorter hub on a stiffer axle will actually free the kart up since it is changing the flex on the ends of the axle, BUT on that same kart, going to a softer axle will tighten the kart up because we're assuming its a softer kart and it would cause too much flex in the rear. Really, if the kart is either too stiff or too soft, it will cause different kinds of tight (maybe one is caused by having a lot of side bite and too much LR lift and the other could be so rigid that there is no lift at the LR at all) but clearly it is not ideal either way.
If properly used, different axle stiffness and different rear hub lengths can be another valuable tuning tool for the karter intent on getting the most out their set-up. But, as with so many things, just buying the pieces and trying them “on the fly” won’t get you very far. The best approach is to devote plenty of dedicated testing time to learn what each change does. You won’t get that kind of time on a race weekend. You’ll have to get out there by yourself when you can commit enough hours to learning what these tools can do for you. Once you have a grip on how each change affects your kart’s handling, then you can use them on race weekends to put your very best package on the grid.
I hope this helps and did not confuse anyone too much.
which is predominantly 1 1/4 axles but.... what the BELL...) Maybe on the lost archives this topic has been hashed (can't recall) but it's & value seems to be rarely tapped discussions seem to be far & few between. So..........Let's discuss all we know about Axelology (flex points, grip, hub length effects, diameter size & length effects on push, grip, over-steer, under-steer, it's effects on chassis stiffness/handling, how it or it's diameter must or must not work with the behind/under the seat type straight/dog-leg cross bar, or how one size axle might work with both American & metric diameter wall thickness tube chassis ) . And perhaps how a in/out board axle clutch effect axle stiffness or should u run a wider hub or wide collar on the opposite side of the axle clutch to equalize stiffness. 
