We dont know how a bicycle works Really?

[Dylan Cram]

I misunderstood what you were getting at. I thought you meant that the round profile of a tire would generate an inwards force, as opposed to generating a steering torque. What I should have posted is that the contact patch on a bicycle with thin tires is too small to generate a significant steering torque, yet such bicycles are very stable due to trail.
A steering torque is indeed induced by the variation of rotational speeds within the contact patch~ given a bike with a front wheel that is free to rotate on a steering axis, and has a circular cross-section. It really cannot be any other way. That is how these things work. That is how three dimensional rotating objects behave. How significant the force is? That seems to be more a matter of personal intuition at this point than anything else. My bike with thin tires has a very short trail relative to other bike designs. It is essentially a track bike, and the design has a short trail and short wheelbase so that it is very responsive to changes in direction. Trail doesn't keep bicycles upright. That has been shown adequately enough for me. The experimental bike that convinced me, however, still had tiny little wheels with a rounded cross-section at contact with the ground.

 

[rcgldr]

A steering torque is indeed induced by the variation of rotational speeds within the contact patch~ given a bike with a front wheel that is free to rotate on a steering axis, and has a circular cross-section. ... How significant the force is? Trail doesn't keep bicycles upright. That has been shown adequately enough for me. The experimental bike that convinced me, however, still had tiny little wheels with a rounded cross-section at contact with the ground.

That experimental bike is supposed to model a two skate bike, one with infinitely thin circular blades instead of wheels (this is why counter rotating wheels were used, to eliminate gyro effects). In the case of that experimental bike, there is no trail / caster, but instead a weight locate above and in front of the front wheel. When the bike leans the weight generates a torque about the vertical axis of the bike (yaw), that causes the front wheel to steer in the direction of the lean. A two skate bike using conventional trail / caster steering geometry would also be self stable.

How significant the force is?

I'm wondering what happens if a conventional bike with trail is let go rider free to travel on a side banked or crowned road? Does it steer sideways uphill because of the different radius at the tire edge versus the tire center, or does it continue to go mostly straight, or does it steer sideways downhill due to the lateral load? If a road were reversed crowned (lowest at the middle), would a rider free bike tend to steer off road, mostly hold it's current line, or tend to steer toward the middle of the road?

update - I tested on a private road with a central drain, where the road is angled somewhere between 5 and 10 degrees on both sides, with an old Centurion Super Lemans with 700c tires. Pushing the bike by the seat and letting it free for a few seconds, there was no tendency to steer "upwards". Due to the small angle, the contact patch would only be slightly offset, so the amount of steering torque related to the offset contact patch was imperceptible. However, leaning the bike on a level surface at around 5 degrees was enough to result in significant inwards steering.

Where I have noticed a steering effect related to an offset contact patch is when braking while leaned over on a motorcycle, which tends to turn the front tire inwards, requiring more counter steering effort to hold a lean angle. The same effect occurs if braking on a banked section of road (the tire tends to steer up the bank). The amount of counter steering effort involved varies between bikes and probably between tire profiles. However the effect is only noticeable under braking. I don't feel any effect on a banked road when not braking.