I worked on this last week while I had things apart for the locker install.
I was curious about the factory scrub radius and how different tire and wheel combinations play into it.
First, I put a magnetic angle finder on top of both ball joints, and took the average of the two to account for any slope in my garage floor or difference between jack stands. The result was 9.3° for the steering axis inclination.
Then, I measured the distance from the steering axis to the wheel mounting surface at 5-1/8". I basically just eyeballed where the axis was, I didn't run a string or anything like that. So this is likely plus or minus part of an inch, but it's less important.
I drew this out, viewed from the front, in AutoCAD, which allowed me to measure the stock scrub radius at 1.8", using a factory Canyon or Ravine wheel which has 5.5" BS, and the rolling radius of a 30x9.5" tire calculated from the revolutions per mile published for a BFG KO2 tire.
From there, it's just trig to calculate the change in scrub radius from the 30" tire package.
View attachment 294069
Step 1: Calculate the scrub radius LOST by the increase in tire radius. This happens because as you can imagine in the image above, the intersection of the contact patch with the steering axis moves out as the tire radius grows. As mentioned above, I like to calculate actual rolling radius based on revolutions per mile. It probably doesn't change the result much if you use manufacturer advertised radius instead, but you want to make sure you use the same method for both tire sizes being compared.
Scrub radius change due to tire size = - change in radius*TAN(Steering Axis Inclination)
*Note that if you do this in Excel, you'll need to multiply the SAI by PI()/180 to use degrees instead of radians.
Using my rig as an example, which has 35 and a rolling radius of 16.75" vs the 14.3" of a 30" tire, this value is -0.4".
Then you add the change in scrub radius due to tire size to the difference in wheel offset. A stock 5.5 BS 15x8 wheel has an offset of 25mm or 1". A 4" BS 15x8 has an offset of -12mm or -0.5". So you subtract the new offset from the original offset, and in this example you get 1.5".
-0.4" + 1.5 " =1.1". So I've added 1.1" to the scrub radius from what it would have had with the 30" tire package for an approximate total of 2.9".
Do the calculation for a stock Rubicon with metric 31's and 5" BS Moab wheels, and you get a +0.42" change from the baseline.
Here's a table showing some common arrangements.
View attachment 294062
I thought it was particularly interesting that a 33" tires can maintain an effectively stock scrub radius using a common 5.25" backspaced wheel - the catch is that you have to use a 10.5" to avoid interference at full lock, and probably with the upper rear spring perch as well. On the other hand, the small backspacing required for 12.5" tires combined with a 33" diameter ends up with a +1.5", or 50% more change vs a 35x12.5. Point for pizza cutters!