OK, you're focused on the shape of the tire when looking at it from the side. That deformation is sidewall deformation, and does not change what you'd measure taking a tape measure around the outside of the tire. The contact patch changes - it gets longer and wider. the longer contact patch is not taken into account in your virtual circle model, but it is still part of the circumference and needs to be accounted for.

The tire doesn't need to stretch or shrink when it deforms - within reasonable limits. And if it doesn't stretch or shrink, the circumference didn't change. All that changed is you no longer have a circle, you have a different closed shape. You can still measure the circumference - which is exactly what the rollout measure does, directly and with no calculations needed. And if the circumference didn't change, the # of rotations needed to go a certain distance haven't changed. All the rest is trying to fit a non-circular curve back into your model for a circle.

If you have some data that shows that deformation due to carrying a load makes a difference in rollout, I'd be very interested to see it.

JB

OK, to simplify:

You claim that what matters is not the circumference, but the radius of some imaginary circle.....that you are calculating from the rollout (which whether you like it or not is by definition the circumference)

You claim that because the radius is different, the rollout will be different.

I say that's all a red herring - calculating the radius is completely unneccesary because we can measure the rollout, which unless the tire is slipping or the tread is somehow shrinking or stretching under very slow rotations is by definition the circumference. (not my definition, it's the definition in every geometry text out there)

Now the real point of contention is whether the circumference can change on a tire within reasonable limits when you place a load on it. I say no, it really can't - at least in any meaningful sense. Not only that but unless folks are very careful the noise from trying to measure rollout will swamp any difference.

You say the effective radius of the virtual circle got smaller - implying that the tire circumference is shrinking because a load is place on it. That seems problematic to me - I say the circumference didn't really change, the contact patch lengthened and that's the piece missing in your model. In reality, I wouldn't be surprised if the tire actually stretched under those conditions, since the rest of the tire would be facing higher effective pressure with the load - but I still think it would be too small to be measured the way most of us are measuring rollout.