A few diagrams always help.
View attachment 80855
In this diagram, the bike is braking at the limit, which as has already been explained upthread, means (unless the road is slippery or you're on a tandem) that the rear wheel is just about to lift of the ground. The rear wheel is now contributing no braking force and the entire braking force comes from the front wheel. F1 is the force exerted by the road on the front tyre by friction and if it was relevant we could work out the deceleration of the bike+rider from F1=ma.
F2 is the force exerted by the forks+steerer assembly to the frame of the bike via the headset bearings. F1 = F2 to a first approximation. Actually, F2 is less than F1 by the ratio of the mass of the front wheel+forks+steerer+stem+bars to the total mass of bike+rider, with a further adjustment for forces exerted on the rider's arms by the bars, but that doesn't change the principle and is likely to be only a few percent, so we can ignore that effect and say F1=F2.
The relation F1=F2 stems from simple mechanics and is true regardless of what happens between F1 and F2. Disk or rim? Makes no difference, F1=F2. Forks flex backwards under the forces? Makes no difference, F1=F2.
If you want a bit more detail, try this diagram.
View attachment 80856
F1 is still the force exerted by the road on the front tyre. But now consider what happens at the headset bearings. At the lower bearing, the steerer is pushing back through the bearing onto the frame. The force exerted on the frame by the steerer is F3. But the whole steerer is being twisted forwards, and at the top bearing, the steerer exerts a forwards force F4 on the frame. The net force on the frame is exactly the same as before: F3-F4=F1.
If we need to, we can work out F3 and F4. Consider now the forces exerted on the forks+steerer:
View attachment 80857
F4 has exactly the same magnitude (action and reaction are equal and opposite) but is in the opposite direction because it’s now the force on the steerer by the frame not the other way round. But now we know that the moments on the steerer+forks are balanced about the lower bearing (because the forks are not undergoing any angular association) so F1.x = F4.y, sufficient information now to work out F3 and F4.
The thing to note is that once again, F3 and F4, just like F2, don’t depend on the type of braking, or the flex of the forks, or anything else like that – just on the geometry and on the limiting value of the braking force F1. (Pedantry qualification: as the fork flexes, then because of the head angle, x would increase slightly and y decrease by an even smaller amount. But this is a small effect. And the flex of the fork doesn’t directly depend on the type of braking either when you’re at the limiting value. It might depend on it indirectly because disk brake forks are likely to be stiffer.)
(all these diagrams consider horizontal components of forces only. Comments about engineers, autistic spectrum, and general nerdishness taken as read, thank you.
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