Would it fall over?
The lean angle on your bike should be sufficient as usually the tire even when flat doesnt completly flatten the sidewall thickness holds up somewhat at least from my experience now if you left it flat for a long time it would probably continue to flatten out. Why do you plan on getting a flat while parked? I have had tires that were way low on air and you would not tell until you put a tire guage on them.
I would say not. There is a fair lean on my bike with it on the stand on flat ground. If the ground is not level and you park off camber you could well have a problem. They are not noted for falling over.
Mine will I think. Depends on the stand and especially if it is lowered. Mine has the SE shocks that lower it about 1 1/2 in the back. I took the SE springs and stand off since to me it looked too low. However, I also park mine with the standing resting on a 2x4 to keep it more upright. I slide a nailed stack of 3 pieces of 2 x 8 x 12" lg under the front edge of the frame. Those tire are mighty stiff however.
In my personal experience, no it didn't fall over. I picked up a screw in my back tire and didn't know it at the time. I was greeted with a totally flat rear the next morning, but the bike was still standing. YMMV.
I've had flats on the front and rear (not simultanously yet) and no problems on the kickstand. I was concerned about it falling over with the front flat while I was riding it at 60 mph. It was fine parked along side the road in the grass though. If you got one of those Breakouts it might just fall over regardless of tire pressure.
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I think it's going to depend on how tall your tires are. Small or low profile tires, not much of a height difference between inflated and flat.
Big 'ol 16" balloon tires, you might have a problem.
Big 'ol 16" balloon tires, you might have a problem.








